Classical Lie algebras

The classical Lie algebras are finite-dimensional Lie algebras over a field which can be classified into four types,, and, where for the General [linear group|general linear Lie algebra] and the identity matrix:

, the special linear Lie algebra;
, the odd orthogonal Lie algebra;
, the symplectic Lie algebra; and
, the even orthogonal Lie algebra.

Except for the low-dimensional cases and, the classical Lie algebras are simple.
The Moyal algebra is an infinite-dimensional Lie algebra that contains all classical Lie algebras as subalgebras.