Maximal Lie subalgebra

A maximal Lie subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ is a proper Lie subalgebra with no intermediate proper Lie subalgebra. A useful sufficient criterion is that the quotient representation $\mathfrak{g}/\mathfrak{h}$ be an irreducible representation under the adjoint action of $\mathfrak{h}$: any intermediate Lie subalgebra gives an invariant subspace of this quotient, hence must be either $\mathfrak{h}$ or $\mathfrak{g}$. For example, $\mathfrak{so}(p)\oplus \mathfrak{so}(q)\subset \mathfrak{so}(p+q)$ has quotient given by the external tensor product of group representations of the two defining modules. For $p,q\ge 3$ these are complex irreducible representations, making the inclusion maximal.