On Lusztig's canonical bases of simple Lie algebras

Let $\mathfrak{g}$ be a simple Lie algebra over~$\mathbb{C}$ with root system~$\mathrm{\Phi }$. In the simply laced case, Frenkel and Kac found a particularly simple construction of~$\mathfrak{g}$, together with a Chevalley basis and explicitly given structure constants, in terms of a certain multiplicative $2$-cocycle $\epsilon \text{\hspace{0.17em}⁣}:\mathbb{Z}\mathrm{\Phi }\times \mathbb{Z}\mathrm{\Phi }\to \{\pm 1\}$. We show that Lusztig’s canonical basis of~$\mathfrak{g}$ can also be obtained in this way, for a suitable choice of~$\epsilon$. We also address the problem of explicitly describing the structure constants when $\mathrm{\Phi }$ is not simply laced.