Diamond Lie algebra
= Diamond Lie algebra
{title2=$\mathfrak d_4$}
The four-dimensional diamond Lie algebra is the <semidirect product of Lie algebras> of the three-dimensional <Heisenberg Lie algebra> with a derivation having eigenvalues zero, one and minus one. With basis $c,d,p,q$, its nonzero brackets are $[d,p]=p$, $[d,q]=-q$, $[p,q]=c$. Its <derived series of a Lie algebra> has dimensions four, three, one and zero. The symmetric form with $B(c,d)=B(p,q)=1$ and all other basis pairings zero is a <nondegenerate bilinear form> and invariant. The alternating tensor $B([x,y],z)$ is the volume form on $\langle d,p,q\rangle$, verifying invariance.