I have a Lie algebra that breaks up into as subalgebra $B$ and its complement $\overline B$ (dividing the set of Lie generators into mutually exclusive subsets $B$ and $\overline B$ such that $B$ is a subalgebra but $\overline B$ is not). Then I find the property: $$ \forall L_a \in B, \forall J_b \in \overline B: $$ $$ [L_a, L_b] \in B, $$ $$ [J_a, J_b] \in B, $$ $$ [L_a, J_b] \in \overline B $$ Here is an example of one instance of this:
[[e1, e2] = -2*e3, [e1, e3] = 2*e2, [e1, e5] = -2*e6, [e1, e6] = 2*e5, [e2, e3] = -2*e1, [e2, e4] = 2*e6, [e2, e6] = -2*e4, [e3, e4] = -2*e5, [e3, e5] = 2*e4, [e4, e5] = 2*e3, [e4, e6] = -2*e2, [e5, e6] = 2*e1]
Does this property have a name? Or is it related to something else I can lookup somewhere? References appreciated.
This Lie algebra $L$ is a perfect $6$-dimensional Lie algebra, i.e., it satisfies $[L,L]=L$. So it must be isomorphic to one of the following Lie algebras: $$ \mathfrak{sl}_2(\Bbb C)\ltimes V(3),\; \mathfrak{sl}_2(\Bbb C)\ltimes \mathfrak{n}_3(\Bbb C),\; \mathfrak{sl}_2(\Bbb C)\times \mathfrak{sl}_2(\Bbb C). $$ Here $V(3)$ is the adjoint $\mathfrak{sl}_2(\Bbb C)$-module, and $\mathfrak{n}_3(\Bbb C)$ is the Heisenberg Lie algebra. The Levi decomposition also yields a vector space decomposition into subalgebras (subspaces which are not subalgebras usually have no name).