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 disentangling identity factors the exponential of a sum of noncommuting Lie-algebra generators into an ordered product of exponentials. For ${\mathfrak{sl}}_2$, a common order is $e^{a{E}_{-}}{e}^{bH}{e}^{c{E}_{+}}$.