= 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 $p,q\ge3$ these are complex <irreducible representations>, making the inclusion maximal.
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