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For this diamond Lie algebra, every bracket lies in $⟨c,p,q⟩$, and each of these three vectors occurs as a bracket. Thus its derived algebra is exactly ${\mathfrak{g}}^{(1)}=⟨c,p,q⟩$. Inside that subalgebra the only nonzero bracket is a multiple of $[p,q]=c$, so ${\mathfrak{g}}^{(2)}=\mathbb{C}c$. Centrality of $c$ then gives ${\mathfrak{g}}^{(3)}=0$. HenceIn particular it is a solvable Lie algebra, although its lower central series of a Lie algebra does not terminate: $[\mathfrak{g},⟨c,p,q⟩]=⟨c,p,q⟩$. Solvability does not imply nilpotence.

If $D:\mathfrak{b}\to \mathrm{Der}(\mathfrak{a})$ 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 $[(a,b),(a^{\prime },b^{\prime })]=([a,a^{\prime }]+D_b{a}^{\prime }-D_{b^{\prime }}a,[b,b^{\prime }])$. 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