 Cartan-Weyl basis

A Cartan-Weyl basis consists of a basis of a Cartan subalgebra together with root vectors $E_{\alpha }$. It makes weight raising and lowering explicit through $[H,E_{\alpha }]=\alpha (H)E_{\alpha }$.

A Chevalley basis imposes an additional integral normalization on these root vectors; that stricter basis is not interchangeable with an arbitrary Cartan-Weyl basis.