I’m currently reviewing some Lie algebra theory, in particular extension of Lie algebras. I’m searching for the name of a certain class of extensions, which generalize central extensions.
Let $\mathfrak{h}$ and $I$ be two Lie algebras. An extension of $\mathfrak{h}$ by $I$ is a short exact sequence $$ 0 \to I \xrightarrow{\;\iota\;} \mathfrak{g} \xrightarrow{\;\pi\;} \mathfrak{h} \to 0 $$ of Lie algebras, i.e. a short exact sequence of vector spaces in which each term is a Lie algebra, and were each map is a homomorphism of Lie algebras. Up to equivalence of extensions we may assume that $$ \mathfrak{g} = \mathfrak{h} \oplus I $$ on the level of vector spaces, that $\iota$ is the canonical inclusion and that $\pi$ is the canonical projection. The Lie bracket of $\mathfrak{g}$ is then of the form $$ [(x,c), (y,d)] = ( [x,y], \kappa(x,y) + \theta(x)(d) - \theta(y)(c) + [c,d] ) $$ for a unique bilinear map $\kappa \colon \mathfrak{h} \times \mathfrak{h} \to I$ and a unique linear map $\theta \colon \mathfrak{h} \to \operatorname{Der}(I)$.
How are those extensions called which arise – up to equivalence – in the case of $\theta = 0$?
If we would also require the Lie algebra $I$ to be abelian then these would be precisely the central extensions of $\mathfrak{h}$ by $I$. Here I’m wondering about the case in which $I$ isn’t required to be abelian.
Moreover, how can these extensions be described without referring to $\kappa$ and $\theta$?
What I have in mind is again the special case of central extension. These can on the one hand be described in terms of $\kappa$ and $\theta$: the Lie algebra $I$ must be abelian, and $\theta = 0$. But they can also be described more intrinsically: the image of $I$ in $\mathfrak{g}$ must be contained in the center of $\mathfrak{g}$.