Diamond Lie algebra (${\mathfrak{d}}_4$)

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 $⟨d,p,q⟩$, verifying invariance.