Splitting and non-splitting extensions in Lie algebras

Question

For Lie algebra $S=\{e_{1}, e_{2}, e_{3}, e_{4}\}$ with non-zero commutations:

$[e_{1}, e_{3}]=e_{1}, [e_{2}, e_{3}]=\alpha\, e_{2}$

we have $S=e_{4}\oplus L_{3}$, such that $L_{3}=\{e_{1}, e_{2}, e_{3}\}$.

My queries is; What are splitting and non-splitting extensions of $e_{4}$? As far I know for splitting extensions of $e_{4}$ I need to find sub-algebras $N$ of $L_{3}$ such that

$[e_{4}, N]\subseteq N$

Am I correct on that? How I am suppose to find these sub-algebras $N$ ? For non-splitting extensions of $e_{4}$, I guess one need to find algebras

$\{e_{4}+\sum_{i}^{3}a_{i}\,e_{i}, N\}$

here again $N$ is sub-algebras of $L_{3}$ such that $\text{Nor}_{S}N$ is not contained in $L_{3}$ (how is that possible !! and why we need so ?) and $e_{4}+\sum_{i}^{3}a_{i}$ is not conjugate to $e_{4}$.

Am I correct on this definition of non-splitting extension of $e_{4}$ ? If yes, how I can find $a_{i}'s$ and sub-algebras $N$ ?

PS. My this query is pertaining to systematic procedure of classifying Lie algebra given in Patera and Winternitz.

Answer 1

A Lie algebra extension $L_2$ of $L_3$ by $L_1$ is a short exact sequence of Lie algebras $$0 \rightarrow L_1\rightarrow L_2\xrightarrow{\pi} L_3\rightarrow 0. $$ The extension is said to be split, if there is a Lie algebra homomorphism $\tau\colon L_3\rightarrow L_2$ such that $\pi\circ \tau=id_{\mid L_3}$. In particular, a semidirect product $L_2\cong L_3\ltimes L_1$ gives a split extension (the direct product $L_2=L_3\oplus L_1$ as well, of course). Equivalence classes of such extensions with abelian Lie algebra $L_1$ are in bijection with the equivalence classes of $2$-cocycles in $H^2(L_3,L_1)$. Here $L_1$ is a $L_3$-module in a natural way. If this second cohomology group is trivial, then all extensions are split.

In your case, $L_2=\langle e_4 \rangle$ is abelian and $1$-dimensional, and $L_3=\langle e_1,e_2,e_3\rangle$. By computing first $H^2(L_3,L_1)$ we will obtain all possible extensions of $L_3$ by $L_1$. Split extensions are semidirect products. For non-split examples, see here, with $L_3=\langle x,y\rangle$ and $[x,y]=x$, $L_1=\langle z\rangle$.