Abelian Complex Structures and Generalizations

After a review on the development of deformation theory of abelian complex structures from both the classical and generalized sense, we propose the concept of semi-abelian generalized complex structure. We present some observations on such structure and illustrate this new concept with a variety of examples.

This author takes the perspective that generalized geometry, especially its deformation theory, is a degree-2 realization of the extended deformation theory developed by Kontsevich et al. [7] [18] [19] [47] [58] [61]. In particular, inspired by the work of Manin and Merkulov [42] [56] [57] [59], this author computed the Frobenius structure on the extended moduli of Kodaira surfaces [61]. One could restrict an analysis of the Frobenius structure from the extended moduli to a generalized moduli in the sense that only degree-2 deformations are allowed. For example, it is proved that the Frobenius structure on the generalized moduli of Kodaira manifolds in all dimensions is trivial [62,Theorem 3].
Lots of geometric consideration and much of the author's work on Frobenius structure and holomorphic Poisson deformation rely on the underlying complex structures being abelian [3] [4] [6] [10] [30] [63] [64]. This class of invariant complex structures was first seen in [9], and is often studied along the line of nilpotent complex structures [28] [29] [69].
Some of the reasons for focusing on abelian complex structures on nilmanifolds M are due to its accessibility in terms of cohomology theory and deformation theory. Deformation theory of generalized complex structures is based on the concept of Lie bialgebroids [40] [49] [50] [51]. Deformation of a generalized complex structure J is controlled by its differential Gerstenhaber algebra DGA(M, J ). Gerstenhaber algebra was invented to study deformation of rings after Kodaira's deformation theory of complex manifolds [35].
Part of this structure is a Schouten bracket on sections of the exterior algebra of a vector bundle [50]. When J is a left-invariant abelian complex structure on a nilmanifold M = ∆\G modeled on a Lie group G, DGA(M, J ) is quasi-isomorphic to its invariant counterpart DGA(g, J ) [17]. The invariant DGA(g, J ) enjoys additional features as stated in Proposition 1 of the next section.
We review the construction of differential Gerstenhaber algebra DGA(M, J ) in Section 3. To motivate our work in Section 4, we present some basic properties when J is a classical object such as complex structure, holomorphic Poisson structure, and symplectic structure.
The goal of this paper is to present the concept of "semi-abelian" generalized complex structures in terms of DGA(g, J ). It is formally stated in Definitions 2 and 3. Definition 2 is made on algebra level. It is designed to capture the features of the invariant part of DGA(M, J ) as if the generalized complex structure J is an abelian complex structure.
It could be extended to analyze generalized complex structures on unimodular algebras [2] [3]. The definition is made so that all abelian complex structures are semi-abelian (generalized) complex structures.
In Section 5, we provide a collection of examples to illustrate the proposed concept.
This collection includes symplectic structure, non-abelian complex structure, and generalized complex structures of various types. In particular we find the existence of semi-abelian (generalized) complex structures that fail to be abelian. There are also nilmanifolds on which there are generalized complex structures but none of them could be semi-abelian.

Kodaira surfaces
A collection of examples of compact complex surfaces have been inspiring objects in complex manifold theory. To name a few, we have complex projective plane, cubic surfaces, K3-surfaces, Hopf surfaces, and Kodaira surfaces. Kodaira surfaces were discovered as a collection of elliptic surfaces in classification of compact complex surfaces [46]. They have trivial canonical bundle. It is also known that the manifold M in question is the co-compact quotient of the complex two-dimensional vector space C 2 and the co-compact lattice transformation leave invariant a complex (2, 0)-form η, subjected to the conditions everywhere on the manifold M . As such Kodaira surface is a nilmanifold with an invariant complex structure.
In 1976, Thurston published his well-known paper presenting Kodaira surface as a non-Kählerian symplectic manifold [71] and remarked that there are an abundance of similar examples in higher dimensions.
Regarding the complex analytic aspect of Kodaira surfaces, Borcea examined the moduli of complex structures on Kodaira surfaces by varying the co-compact lattices of transformations [13]. The starting point of his computation is to recognize that the algebra g on the underlying real vector space for C 2 is solely given by All other Lie brackets are equal to zero. Therefore, as a Lie algebra it is the direct sum of the trivial one-dimensional algebra and the real three-dimensional Heisenberg algebra h 3 .
Borcea's computation of the moduli relied on parametrization of the complex 2-form η in (1) with respect to the basis of the algebra {X 1 , X 2 , X 3 , X 4 } and its dual.

Nilpotent complex structures
On any compact complex surface, the Frölicher spectral sequence associated to the Dolbeault bicomplex degenerates at first step [11] [46]. In a cluster of papers [ A compact manifold M is a nilmanifold if it is the quotient of connected simplyconnected nilpotent Lie group G by a discrete subgroup ∆ so that M = ∆\G [54]. It is a fundamental observation that such a Lie group gives rise to a nilmanifold if and only if its Lie algebra g admits a basis with respect to which the structure constants are rational [54]. An invariant complex structure on a nilmanifold M as a right quotient space is given by a left-invariant complex structure on the Lie group G. Equivalently, it is a real linear map J : g → g such that J • J = −1 and its Nijenhuis tensor vanishes. i.e., for all X, Y in One of the discoveries by Cordero et al. was the concept of nilpotent complex structures, see [26] as a preprint reference for [23]. This subject was further developed by Salamon [69], which has become a key reference for this subject for many publications including this one.
Given an invariant complex structure J on M = ∆\G, the space g 1,0 of (+i)-eigenvectors for J in complexified Lie algebra g C forms a complex Lie algebra. Denote the (−i)eigenspace by g 0,1 . The dual spaces are respectively g * (1,0) and g * (0,1) . The complex structure J is nilpotent if there exists an ordered basis {ω 1 , . . . , ω m } for g * (1,0) and constants A j kl and B j kl such that for all 1 ≤ j ≤ m, In the meantime, inspired by a search for geometry with finite holonomy, a class of complex structures on nilmanifolds emerged in [9]. It satisfies the condition that for all This condition is equivalent to require the (+i)-eigenspace g 1,0 to form a complex abelian algebra although g and g C are not necessarily abelian. Such complex structures are called abelian complex structures. In [69] it is proved that if a nilpotent algebra g admits an abelian complex structure then there exists an ordered basis {ω 1 , . . . , ω m } for g * (1,0) and constants A j kl such that for all 1 ≤ j ≤ m, We address this basis for g * (0,1) an ascending basis. The observations above lead to the next proposition.
Proposition 1 Let J be a left-invariant complex structure on a nilmanifold with Lie algebra g. Let g 1,0 be the space of invariant (1, 0)-vectors and g * (0,1) the space of invariant (0, 1)-forms. The following conditions are equivalent.
J is abelian if and only if one of these conditions is satisfied.
Low-dimension nilpotent algebras are classified in [37]. Given the choice of basis and constraints in (4) for nilpotent complex structures and those in (6) for abelian complex structures, there is a classification of the underlying nilpotent algebras to admit such complex structures when the real dimension of the algebra is at most six [26] [27] [69].
A coarse dimension count on family of invariant complex structures on each admissible algebra was also done in [69]. Classification of abelian complex structures in low dimension is also extended to algebras other than the nilpotent ones [2].
Notations. To further present our work, we adopt a convention popularized by [69].
Suppose that e j , e k , e l are 1-forms we use e jkl to represent their exterior product e j ∧ e k ∧ e l . Likewise when e j , e k are vectors, the bivector e j ∧ e k is represented by e jk . When {e 1 , . . . , e n } is an ordered basis for g * , the structure equations for the algebra g in terms of the Chevalley-Eilenberg differential is represented by the n-tuple (de 1 , . . . , de n ). For example, when de n = e ab + e kl the last entry in (de 1 , . . . , de n ) will be represented by ab + kl.
With the above notations, the algebra of the real 3-dimensional Heisenberg algebra h 3 is represented by (0, 0, 12) and the non-trivial algebra with invariant complex structure given in (2) is isomorphic to (0, 0, 0, 12) = R ⊕ h 3 .

Examples of abelian complex structures
There are only five non-trivial six-dimensional 2-step nilpotent algebras admitting abelian complex structures [27] [69]. Each has a high-dimension generalization.
After a change of bases e 4 to −e 4 , e 5 to −e 6 and e 6 to e 5 , we present the same algebra with structure equations below.

Cohomology and deformation
Given a nilmanifold M = ∆\G, there is an inclusion of left-invariant differential forms in the space of smooth differential forms.
Nomizu proved that this inclusion is a quasi-isomorphism in the sense that the inclusion map induces an isomorphism at cohomology level [60], where H k (g) is the k-th cohomology with respect to the complex of the Chevalley-Eilenberg differential of g. In other words, for any ω ∈ g * and X, Since the attempt by Sakane on similar quasi-isomorphism for Dolbeault cohomology [68], it has been proved that when M is a nilmanifold with a nilpotent complex structure one obtains a natural quasi-isomorphism.
Theorem 1 [26] Suppose that M = ∆\G is a nilmanifold with a nilpotent complex structure, the inclusion of invariant (p, q)-forms in the space of sections of (p, q)-forms is a quasi-isomorphism. i.e., the inclusion map induces an isomorphism at cohomology level: In addition, Console and Fino initiated a study on the same issue for all invariant complex structures from the perspective of stability of the desired quasi-isomorphisms [20] [21]. It is remarkable that the above statement remains an open conjecture when the complex structure is merely invariant. For advancement in this direction, please see [66] [67] and the latest development in [33] and references therein.
This author and collaborators took on the issue of deformation of invariant complex structures on nilmanifolds in [52] with a focus on 2-step nilmanifolds with abelian complex structures. The key to enable this investigation was a result similar to Theorem 1. When Θ is the sheaf of germs of holomorphic tangents for the complex nilmanifold M , the result states as below.
Theorem 2 Suppose that M = ∆\G is a nilmanifold with an abelian complex structure. The inclusion map induces an isomorphism at cohomology level: The above theorem was initially proved on 2-step nilmanifolds [52,Theorem 1]. It is subsequently expanded to include nilmanifolds with arbitrary number of steps [22, Theorem 3.6]. Both [22] and [52] rely on the fact that the center of the algebra g is invariant of the complex structure J when it is abelian. It recreates a (series of) holomorphic fibrations with complex torus as fibers. The proof of Theorem 2 becomes an application of Leray spectral sequence formalism.
The quasi-isomorphism in Theorem 2 enables a construction of moduli of complex structures on Kodaira manifolds in [39] along the line of Borcea's work in [13]. It also enables an analysis on the stability of abelian complex structures under deformation [22].
More broadly, in [52] when an abelian complex structure J is given on a 2-step nilmanifold, the authors considered the Kuranishi space Kur(J) of the given complex structure J and the subspace Abel(J) consisting of local deformation parameter space of abelian complex structures with J in the center. Among other results, they found the following.

Generalized Complex Structures
A generalized complex structure could be conceived as both a tensorial object and a spinorial object subjected to multiple conditions. Its investigation was initiated by Hitchin [43] and developed by Gualtieri [40]. Given any point x on the manifold M , when X, Y are in T x M and α, β in T * x M , one considers the pairing between two elements of T An almost generalized complex structure is a bundle map where B is a two-form and Π is a bivector [40]. Equivalently, the bundle of (+i)-eigenspace L is maximally isotropic subbundle of the complexification of T M ⊕ T * M . Let L represent the complex conjugate bundle of L, it is the bundle of (−i)-eigenspace. By virtual of L being maximally isotropic, L is also maximally isotropic and Since L and L are maximally isotropic and the bilinear form −, − is non-degenerate, it yields naturally isomorphisms by the pairing −, − as given in (11): In subsequent discussion, we often utilize these isomorphisms.
The Courant bracket defined on the space of sections where L X β is the Lie derivative of the one-form β along the vector field

DGA and cohomology
Under the assumption that a generalized complex structure J is integrable, one treats L as a Lie algebroid [40] [49] and extends the restriction of the Courant bracket to the space of section of exterior algebras C ∞ (M, ∧ • L) to obtain what is known as a Gerstenhaber algebra, also as a Schouten algebra [35] [50, Definition 7.5.1]. We will address the restriction of the Courant bracket on C ∞ (M, ∧ • L) as the Schouten bracket.
Taking complex conjugations, the space C ∞ (M, ∧ • L) also inherits a Schouten bracket.
Since L is a Lie algebroid, it has an associated differential on its dual L * ∼ = L: It is observed in [40] that the pair (L, L) forms a Lie bialgebroid in the sense of [51]. The operator δ extends to an even exterior differential operator and an odd differential Lie In summary, each generalized complex structure is associated to a differential Gerstenhaber algebra [35] [61].
It is further observed in [40] that the operator δ is elliptic and the integrability of J is equivalent to δ • δ = 0. Therefore DGA(M, J ) has an associated cohomology theory .
As a consequence of (17), the exterior product and the Schouten bracket descend to the cohomology space so that we obtain a Gerstenhaber algebra: We obtain a generalized deformation when Γ is a degree-2 section, i.e., an element in

Symplectic manifolds
Suppose that M is a smooth manifold and Ω is a symplectic form, we treat a contraction of Ω with a tangent vector as a bundle map Ω : Since Ω is non-degenerate, its inverse Ω −1 is a well-defined bivector. In the matrix representation of a generalized complex structure J , by choosing J = 0, B = Ω, and Π = Ω −1 , one obtains a generalized complex structure. Pointwisely, The integrability of J as a generalized complex structure amounts to the identity for all X, Y in C ∞ (M, T M ). It is equivalent to dΩ = 0. Let X, Y, Z be any smooth vector Consider a bundle map obtained by a composition of inclusion and projection. ϕ is a bundle isomorphism because Ω is non-degenerate. It extends naturally to a bundle map of exterior powers so that for where the bracket on the right hand side is the usual Lie bracket of vector fields. By (21), ] Ω . In particular, the differential Gerstenhaber algebra DGA(M, Ω) •]] is isomorphic to the differential Gerstenhaber algebra defined on the space of differential forms where the differential d is the deRham differential. The resulting cohomology is the com-

Proposition 3
When Ω is a symplectic form, the bundle map (23) defines an isomorphism of differential Gerstenhaber algebra.
The isomorphism above demonstrates that the differential Gerstenhaber algebra of a symplectic structure as a generalized complex structure is consistent with the ones in extended deformation theory in [

Classical complex structures
The matrix representation of J in (13) for a classical complex structure is given by B = 0 and Π = 0. The bundles of (+i) and (−i) eigenvectors with respect to J are respectively where T 1,0 and T 0,1 are bundles of (1,0)-vectors and (0,1)-vectors with respect to J. T * (1,0) and T * (0,1) are their respective duals. As a result, it is now also obvious that in terms of the ascending basis and its dual

Holomorphic Poisson structures
Holomorphic Poisson structure is built upon a complex structure J. In a matrix repre- The (+i)-eigenbundle L with respect to J has a real representation in terms of Π and a complex representation in terms of Λ.
It is most convenient to express the integrability of L in terms of the second representation because its integrability is equivalent to J being an integrable classical complex structure and Λ is a holomorphic Poisson structure, i.e., ∂Λ = 0 and [[Λ, Λ]] = 0.
The last is equal to zero for all ω and all V if and only if ∂U = 0 [34]. Next for any (1,0)-forms µ, ν, It is equal to zero for all µ, ν if and only if L U Λ = 0. As the complex structure is integrable, the type-(0,2) component of dν vanishes. Therefore, Next, for any (1,0)-forms µ, ν,

Different types of generalized complex structures
There are many examples of generalized complex structures different from the symplectic or complex types. For example, one could have a symplectic bundle over complex manifold [4]. On the total space of such a fiber bundle, there exists a closed 2-form such that its restriction to each fiber is a symplectic form. The base manifold is a complex manifold.
Below we apply a well-known method to construct a generalized complex structure on a manifold that is neither symplectic type or complex type. Let U (1) represent the onedimensional unitary group.
Proposition 5 Suppose that M is the total space of a principal U (1) × U (1)-bundle over a complex n-dimensional manifold X . If the curvature of a connection on this bundle is represented by type-(1,1) forms on X, then M admits a generalized complex structure of type-n on the manifold M .
Proof: We prove this theorem by mimicking a well-known construction of integrable complex structure on toric bundles. See e.g., [38].
Let V be the bundle of vertical vector fields generated by the principal action of U (1)×U (1).
It yields an exact sequence of vector bundles on the manifold M : Let H = ker θ = ker θ 1 ∩ ker θ 2 be the horizontal space of the connection θ.
Therefore, we obtain an almost generalized complex structure J with (+i)-eigenbundles given by It is apparent that L is isotropic with respect to the non-degenerate pairing (11). The If u, u 1 , u 2 are (1,0)-vector fields on the base manifold X, u h 1 and u h 2 are (+i)-eigenvectors with respect to J on the manifold M . On M , because ϑ 1 and ϑ 2 are both type-(1,1) forms.

On the other hand, a vertical (+i)-eigenvector is given by
Since all horizontal distributions are invariant of the principal action and u h is in ker θ 1 ∩ ker θ 2 , Since ϑ 2 is type-(1,1) and u is a type-(1,0), ι u ϑ 2 is a type-(0,1) form on the manifold X.
It follows that last term is a section of π * T * (0,1) X ⊂ L. If ω is a (0,1)-form on the base complex manifold X, Since the pull-back form π * ω is invariant of vertical vector field, the above is equal to zero.
. Therefore, the space of sections for L is closed with respect to the Courant bracket.
There are lots examples of non-trivial toric bundles as described by Proposition 5 with very interesting complex geometry; see e.g., [38]. In this note, we illustrate the above construction with Example 7.

Deformation
Recall that the construction for the differential Gerstenhaber algebra The pair forms a new Lie bialgebroid, or equivalently new generalized complex structure in our context if and only if Γ satisfies the Maurer-Cartan equation (19) and the pair stays maximally isotropic [40] [49]. Furthermore, the direct sum L Γ ⊕ L Γ has alternative representation when Γ is sufficiently close to zero. Theorem 3 [49] The differential δ on L as a dual to L and the differential δ Γ on L as a dual to L Γ are related by for all section a in C ∞ (M, ∧ • L). In particular, DGA(M, J Γ ) after deformation by Γ is From now on, we denote [[Γ, a]] by ad Γ (a). In the notations above, the Maurer-Cartan equation (19) is translated to δ Γ • δ Γ = 0 [49]. The next corollary is trivial. When a generalized complex structure J is given by a holomorphic Poisson structure (J, Λ), from the viewpoint of Theorem 3 we treat Λ as a deformation of the complex structure J. In terms of a matrix representation with components J and Π, we have where t is the deformation parameter. When t = 0, we reach the underlying complex structure. When t = 1, we have our holomorphic Poisson structure. In this perspective, the algebroid differential δ with respect to the generalized complex structure associated to (J, Λ) is identified to ∂ + ad Λ acting on the Lie algebroid of the complex structure J, and Lemma 1 and Lemma 2 follow easily.
In particular, the non-zero Courant brackets on g C are Since [[T j , ω k ]] = ι T j dω k , the non-zero brackets between elements in g 1,0 and elements in Finally, as for any T ∈ g 1,0 , Therefore, Λ = T 2 ∧ T 3 is a holomorphic Poisson structure. By structure equations

Nilmanifolds
Suppose that M is a nilmanifold M = ∆\G and the generalized complex structure J is invariant, there is an inclusion of left-invariant sections in the space of smooth sections for various bundles.
Since the evaluation of invariant forms on invariant vectors are constants, the restriction of the Courant bracket to g ⊕ g * is reduced to If ℓ and ℓ represent the space of invariant sections for the Lie bialgebroid L and L respectively, we have ℓ ⊕ ℓ = (g ⊕ g * ) C and the inclusions Since the differential δ on C ∞ (M, ∧ k L) is due to the invariant Lie algebroid structure on C ∞ (M, L), the inclusion map ι of ℓ intertwines with the differential δ on ℓ so that we have an inclusion of invariant differential Gerstenhaber algebra: Problem 1 When will the inclusion map be a quasi-isomorphism?
In view of Proposition 3 and Nomizu's Theorem [60], when the generalized complex structure is a symplectic structure, we have an isomorphism. As noted in previous sections, the work regarding generalizing Nomizu's work to Dolbeault cohomology on complex structure on nilmanifolds remains an on-going effort in the past twenty years.

Semi-Abelian Generalized Complex Structure
We are interested in exploring ways to extend the concept of abelian complex structure to generalized complex manifolds. Consider the definition of abelian complex structure as given in (5). Let J be a generalized complex structure. One may attempt to expand it naïvely so that for all X, Y ∈ g and α, β ∈ g * , However, when J is a symplectic structure Ω it forces the algebra g to be abelian because we would have Yet the Courant bracket between a pair of 1-forms is equal to zero.
As our goal is to create a theory to adopt to cohomological computation to facilitate investigation deformation of generalized complex structures, we focus on the structure of the differential Gerstenhaber algebra for a generalized complex structure as given in (18).
Let us once again recall the key features of a classical complex structure J being abelian is presented in the two equivalent conditions in Proposition 1. This proposition and our need in computing cohomology effectively at least in some situation drives our proposed concept below.
Let G be the vector space g ⊕ g * equipped with the Courant bracket induced by the Lie bracket on g. It is a Lie algebra with the properties that g is a subalgebra and g * is an abelian ideal. Therefore, G is a semi-direct product G = g ⋊ g * .
Definition 1 Suppose that A is a subalgebra of G = g ⋊ g * , K is an abelian ideal of G, and both A and K are maximally isotropic with respect to the natural pairing (11), the semi-direct product presentation G = A ⋊ K is called admissible. We will also call (A, K) an admissible pair associated to the Lie algebra g.
By virtual of being maximally isotropic, the non-degenerate pairing defines an isomor- Since the restriction of J on A satisfies Identity (5), a is an abelian complex algebra.
Therefore, ℓ is a semi-direct product a ⋊ k of two abelian subalgebras. As A ∼ = K * by the natural non-degenerate pairing, Since for any element ℓ 1 ∈ ℓ, δℓ 1 is obtained by the evaluation on every pair of elements ℓ 2 , ℓ 3 in ℓ: Given the structure of ℓ = a ⋊ k and a and k are abelian, [[ℓ, ℓ]] ⊆ k. By (43), we find that k is in the kernel of δ and the image of a via δ is contained in a ⊗ k.
Proposition 6 Let J be a semi-abelian complex structure on a Lie algebra g with admissible pair (A, K), the following holds.
• a is abelian subalgebra and k is an abelian ideal such that ℓ = a ⋊ k.
The last point makes obvious restriction on the DGA(g, J ) in terms of a lower bound on the dimension of its first cohomology H 1 (g, J ). This proposition mirrors the observation on abelian complex structures as noted in Proposition 1. Next, we find constraints on symplectic structure being semi-abelian.
Proposition 7 Suppose that Ω is a symplectic structure on a nilpotent algebra g. If (A, K) is Ω-admissible, there exist an abelian subalgebra b in g and an abelian ideal h in g such with dΩ(X) = 0 for all X ∈ h. In particular, g is a semi-direct product g = b ⋊ h.

Proof:
When Ω is a semi-abelian symplectic form, suppose that ℓ = a ⋊ k. Then for some vector subspaces b and h in g. For any X, Y ∈ g, Therefore We may now restrict the scope of Problem 1 to our current perspective. Problem 3 Suppose that J is a semi-abelian generalized complex structure. Find all Γ in ∧ 2 ℓ such that (L Γ , L Γ ) is a semi-abelian generalized complex structure.

Remark 1
The concept of semi-abelian generalized complex structure as given in Definition 2 on algebra level potentially could be extended in a context similar to those in [2] [8] [48].

Examples
By construction, all abelian complex structures are semi-abelian (generalized) complex structures. In this section, we present a collection of examples of semi-abelian generalized complex structures, including symplectic structures, non-abelian nilpotent complex structures, together with non-complex and non-symplectic type structures.
If a symplectic structure is semi-abelian as a generalized complex structure, we will address it as a semi-abelian symplectic structure. In Example 10, we find a non-abelian complex structure such that as a generalized complex structure, it is semi-abelian. In short, there are semi-abelian complex structures that fail to be abelian. We start on four-dimension cases.  Therefore, J is integrable and semi-abelian.
Example 6 An example admitting no semi-abelian generalized complex structures.
Consider the algebra (0, 0, 12, 13). Since de 3 = e 12 , de 4 = e 13 , it admits symplectic structures such as e 23 + e 14 . However, this algebra does allow a semi-direct product as prescribed by Proposition 7, there is no semi-abelian symplectic structure on this algebra.  It follows that the (+i)-eigenspace is spanned as Therefore, the generalized complex structure J is not semi-abelian. does not admit any invariant complex structure, let alone a semi-abelian one.
As the structure for Lie algebra g is de 6 = e 12 + e 34 , the structural equations associated to G = g ⋊ g * are As seen in Example 1, (g, g * ) forms an admissible pair for an abelian complex structure.
On the other hand, the pair below is also admissible.
The structure equations for this G = g ⋊ g * are given below.