Lie subalgebra

Inside a big symmetry algebra there are often smaller self-contained worlds — collections of infinitesimal symmetries that close up among themselves and never escape. A Lie subalgebra is such a closed-off piece: a subspace where bracketing any two of its members never takes you outside it. It is the Lie-theoretic analogue of a subgroup or a subring.

Formally, a subspace M of a Lie algebra L is a Lie subalgebra if it is closed under the bracket: [x, y] lies in M whenever x, y lie in M. Since the bracket is already bilinear, antisymmetric, and Jacobi on all of L, these properties are inherited automatically, so M is a Lie algebra in its own right. The only thing to check is closure.

Subalgebras need not be ideals: closure under bracketing among elements of M is weaker than the requirement [M, L] inside M. For instance the diagonal trace-zero matrices form an abelian subalgebra of sl(n) (a Cartan subalgebra) which is not an ideal — bracketing a diagonal matrix with an off-diagonal one lands outside the diagonals.