In this section we briefly list functions for creating other structures from a reflection group. See the appropriate chapters of the Handbook for more details.
The Coxeter group isomorphic to the real reflection group W. See Chapter COXETER GROUPS.
The permutation Coxeter group isomorphic to the finite real reflection group W. See Chapter COXETER GROUPS.
The reductive Lie algebraover the ring R with Weyl group W. Unless W is finite, real, and crystallographic, an error is flagged. See Section Almost Reductive Lie Algebras.
The group of Lie typeover the field k with Weyl group W. Unless W is finite, real, and crystallographic, an error is flagged. The roots and coroots of W must also have integral components. See Chapter GROUPS OF LIE TYPE.