  For a semisimple Lie algebra over the complex numbers, Dynkin (1952)
developed an algorithm to classify the regular semisimple subalgebras, up to
conjugacy by the inner automorphism group. For a graded semisimple Lie algebra
over the complex numbers, Vinberg (1979) showed that a classification of a
certain type of regular subalgebras (called carrier algebras) yields a
classification of the nilpotent orbits in a homogeneous component of that Lie
algebra. Here we consider these problems for (graded) semisimple Lie algebras
over the real numbers. First, we describe an algorithm to classify the regular
semisimple subalgebras of a real semisimple Lie algebra. This also yields an
algorithm for listing, up to conjugacy, the carrier algebras in a real graded
semisimple real algebra. We then discuss what needs to be done to obtain a
classification of the nilpotent orbits from that; such classifications have
applications in differential geometry and theoretical physics. Our algorithms
are implemented in the language of the computer algebra system GAP, using our
package CoReLG; we report on example computations.
