= Toral subalgebra
A toral subalgebra of a <Lie algebra> $\mathfrak g$ is a <Lie subalgebra> $\mathfrak a$ such that $\operatorname{ad}a$ is <diagonalizable> for every $a\in\mathfrak a$. Here <diagonalizability> is over the stated ground <field>, and refers to the <Adjoint representation>. In arguments that use a simultaneous <weight decomposition>, commutativity of $\mathfrak a$ is stated as well. For a <semisimple Lie algebra> over an <algebraically closed field> of <characteristic zero>, <Cartan subalgebras> are examples. Semisimplicity on the <defining representation> of a <symplectic Lie algebra> follows from <adjoint semisimplicity in the symplectic Lie algebra>.
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