  Given a grading on a nonassociative algebra by an abelian group, we have two
subgroups of automorphisms attached to it: the automorphisms that stabilize
each homogeneous component (as a subspace) and the automorphisms that permute
the components. By the Weyl group of the grading we mean the quotient of the
latter subgroup by the former. In the case of a Cartan decomposition of a
semisimple complex Lie algebra, this is the automorphism group of the root
system, i.e., the so-called extended Weyl group. A grading is called fine if it
cannot be refined. We compute the Weyl groups of all fine gradings on simple
Lie algebras of types A, B, C and D (except D4) over an algebraically closed
field of characteristic different from 2.
