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The classical Lie algebras are finite-dimensional Lie algebras over a field which can be classified into four types A n {\displaystyle A_{n}} , B n {\displaystyle B_{n}} , C n {\displaystyle C_{n}} and D n {\displaystyle D_{n}} , where for g l {\displaystyle {\mathfrak {gl}}} the general linear Lie algebra and I n {\displaystyle I_{n}} the n × n {\displaystyle n\times n} identity matrix:
Except for the low-dimensional cases D 1 = s o {\displaystyle D_{1}={\mathfrak {so}}} and D 2 = s o {\displaystyle D_{2}={\mathfrak {so}}} , the classical Lie algebras are simple.
The Moyal algebra is an infinite-dimensional Lie algebra that contains all classical Lie algebras as subalgebras.