Cartan-Weyl basis
= Cartan-Weyl basis
{c}
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.