A Table of Lie Groups and Lie Algebras

Type	Radical (maximal solvable ideal)	Levi subalgebra (semisimple subalgebra complementing the radical)	Cartan subalgebra (self-normalizing nilpotent subalgebra) Maximal toral subalgebra (maximal abelian subalgebra of semisimple elements)	Borel subalgebra (maximial solvable subalgebra)
\(\mathfrak{gl}(n)\), all matrices	scalar matrices	\(\mathfrak{sl}(n)\)	(not semisimple)
\(\mathfrak{sl}(n)\), matrices with trace \(0\)	(simple)		diagonal matrices with trace 0	upper triangular matrices with trace \(0\)
\(\mathfrak{so}(n)\), skew-symmetric matrices	(semisimple)		block-diagonal …
\(\mathfrak{sp}(2n)\)	(simple)		\(\operatorname{diag}(h_1,\dots,h_n,−h_1,\dots,−h_n)\)