The metric tensor B m n on the Lie group is the Cartan metric, aka the Killing form. Note that, as a matrix, the second W is the transpose. For N a (pseudo-) Riemannian manifold, the metric is a (pseudo-) Riemannian metric. The above generalizes to the case of symmetric spaces. These vielbeins are used to perform calculations in sigma models, of which the supergravity theories are a special case.