Weight decomposition of sld( R) with respect to the adjoint representation of so(p,q)

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