Weight decomposition of ${\mathfrak{s}\mathfrak{l}}_d(\mathbb{R})$ with respect to the adjoint representation of $\mathfrak{s}\mathfrak{o}(p,q)$

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