= Semidirect product of Lie algebras
{title2=$\mathfrak a\rtimes\mathfrak b$}
If $D:\mathfrak b\to\operatorname{Der}(\mathfrak a)$ is a <Lie algebra homomorphism> into the <derivations of a Lie algebra>, the semidirect product has underlying vector space $\mathfrak a\oplus\mathfrak b$ and bracket $[(a,b),(a',b')]=([a,a']+D_ba'-D_{b'}a,[b,b'])$. The homomorphism and derivation identities ensure the <Jacobi identity>. The first factor is an ideal; the second is a subalgebra whose action records the twisting. The <diamond Lie algebra> arises by letting a one-dimensional algebra act on a <Heisenberg Lie algebra>.
// Target: linear-algebra.bigb
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