<-- Go Back Last Updated: 11/01/2025

Abelian Subalgebra**

Given a Lie algebra \(\mathfrak{g}\), a subalgebra \(\mathfrak{h}\subseteq\mathfrak{g}\) is called abelian if it has a trivial bracket with itself: \(\forall x,y\in\mathfrak{h}:[x,y]=0\). Equivalently, the subalgebra product of \(\mathfrak{h}\) with itself is trivial, \([\mathfrak{h},\mathfrak{h}]=0\). If all of \(\mathfrak{g}\) is an abelian subalgebra, we call \(\mathfrak{g}\) abelian.