Using the notion of a graph with compatible automorphism, Lusztig constructs all symmetrizable Cartan data (i.e. Cartan matrices $A$ for which there is a diagonal matrix $D=\mathrm{diag}(d_1,\ldots,d_n)$ with integer entries such that $DA$ is symmetric).

I am wondering if anyone knows a natural condition on this construction so that $\gcd(d_1,\ldots,d_n)=1$?