Let g be a semi-simple Lie algebra and h a Cartan subalgebra.  Let be a basis forThe linear transformations are simultaneously diagonalizable over C -- if  x 2 g is a common eigenvector for all these transformations, then  .  The -tuples   2   called the roots Δ of  with respect to the Cartan subalgebra .  The eigenspace decomposition is called the root space decomposition of g with respect to h.  

For each root and corresponding root space , this query checks that .  It also checks that the span of the Cartan subalgebra and the root spaces is the full Lie algebra  .

With output = "root", this query will return the root  if the equations fail.