Real forms of embeddings of maximal reductive subalgebras of the complex simple Lie algebras of rank up to 8

We consider the problem of determining the noncompact real forms of maximal reductive subalgebras of complex simple Lie algebras. We briefly describe two algorithms for this purpose that are taken from the literature. We discuss applications in theoretical physics of these embeddings. The supplementary material to this paper contains the tables of embeddings that we have obtained for all real forms of the semisimple Lie algebras of rank up to 8.