Computing the real Weyl group

Let g be a semisimple Lie algebra over the field of real numbers. Let G be a real Lie group with Lie algebra g. The real Weyl group of G with respect to a Cartan subalgebra h of g is defined as W(G,h)=NG(h)/ZG(h). We describe an explicit construction of W(G,h) for Lie groups G that arise as the set of real points of connected algebraic groups. We show that this also gives a construction of W(G,h) when G is the adjoint group of g. This algorithm is important for the classification of regular semisimple subalgebras, real carrier algebras, and real nilpotent orbits associated with g; the latter have various applications in theoretical physics.