Derived subalgebra is an ideal

Let $\mathfrak g$ be a Lie algebra .

Lemma

The derived subalgebra $[\mathfrak g,\mathfrak g]$ is an ideal in $\mathfrak g$; equivalently,

Proof

Take $x\in\mathfrak g$ and an element of $[\mathfrak g,\mathfrak g]$ of the form $[y,z]$. Using the Jacobi identity for the Lie bracket ,

Each term on the right is a commutator of two elements of $\mathfrak g$, hence lies in $[\mathfrak g,\mathfrak g]$. By linearity, $[x,w]\in[\mathfrak g,\mathfrak g]$ for all $w\in[\mathfrak g,\mathfrak g]$, proving the claim.

Context. This lemma ensures that iterating $[\cdot,\cdot]$ produces characteristic ideals (e.g. the derived series ), making solvability a robust isomorphism-invariant notion.