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 Lie algebra and the identity matrix:
- , the special linear Lie algebra;
- , the odd-dimensional orthogonal Lie algebra;
- , the symplectic Lie algebra; and
- , the even-dimensional 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.