Let g be a Lie algebra and h a Cartan subalgebra. Let be a basis for . A root for g with respect to this basis is a non-zero -tuple of complex numbers such that    (*)  for some .

The set of  which satisfy (*) is called the root space of g defined by and denoted by  A basic theorem in the structure theory of semi-simple Lie algebras asserts that the root spaces  are 1-dimensional.

The first calling sequence calculates the root  for the given root space . If is not a root space, then an empty vector is returned.

The second calling sequence simply returns the indices, as column vectors, for the table defining the root space decomposition.