Geodesic
Weyl Groups of Fine Gradings on Simple Lie Algebras of Types A, B, C and D
Serdica Mathematical Journal, Tome 38 (2012) no. 1-3, pp. 7-36.

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Given a grading Γ : L ⨁ = g ∈ G L g on a nonassociative algebra L by an abelian group G, we have two subgroups of Aut(L): the automorphisms that stabilize each component L g (as a subspace) and the automorphisms that permute the components. By the Weyl group of Γ 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 D 4) over an algebraically closed field of characteristic different from 2.
Keywords: Graded Algebra, Fine Grading, Weyl Group, Simple Lie Algebra 
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Elduque, Alberto; Kochetov, Mikhail. Weyl Groups of Fine Gradings on Simple Lie Algebras of Types A, B, C and D. Serdica Mathematical Journal, Tome 38 (2012) no. 1-3, pp. 7-36. http://geodesic.mathdoc.fr/item/SMJ2_2012_38_1-3_a1/