I am trying to understand some construction done by Lusztig in his book on quantum groups. Given some Cartan datum, let $U=U_q(\mathfrak{g})$ the standard quantized enveloping algebra of the Kac-Moody algebra $\mathfrak{g}$. Its negative part $U^{-}$ (the subalgebra of $U$ generated by the $f_i$'s) has a canonical basis $B$. In his book Introduction to Quantum Groups (don't be fooled by the title!), Lustzig constructs the non-unital extension $\dot{U}$ and proves that it has a canonical basis $\dot{B}$.

As a set, $\dot{B}$ is in bijection with $B\times X \times B$ (where $X$ is the lattice of weights, normally called $P$ in any other books on quantum groups), and its elements are described with the rather cryptic notation $b\diamond_\zeta b''$. The definition of those elements is however very obscure and non-explicit. I would appreciate finding an easier, more explicit, description, like the one given in section 25.3 in Lustzig's book for $U_q(\mathfrak{sl}_2)$, also described in great detail by Lauda in the first part of his paper A categorification of quantum sl(2). It is not clear to me how this description can be extended to more complicated quantum groups.

Does anyone know any similar simple description of the canonical basis $\dot{B}$, even in some other particular cases? I am also very interested of knowing if is there any relation with crystals, akin to the equivalence between $B$ and Kashiwara's crystal basis $B(\infty)$.