  In this concise article, we compute the weight decomposition of
$\mathfrak{sl}_d(\mathbb R)$ with respect to the adjoint representation of
$\mathfrak{so}(p,q)$, where $d=p+q$ and demonstrate in detail that
$\mathfrak{sl}_d(\mathbb R)$ comprises two irreducible
$\mathfrak{so}(p,q)$-invariant subspaces. This can be employed to establish the
well-known fact that the identity component of $\mathrm{SO}(p,q)$ is a maximal
connected subgroup of $\mathrm{SL}_d(\mathbb R)$.
