The $H$s are always going to be given with the help of roots themselves, I think.
They are just diagonal with the diagonal elements given by $a^{(i)}_j$ for the $i$th Cartan generator Cartan generator $H^i$ where $a^{(i)}_j$ is the $j$th component of the $i$th root vector and $i=1,...,\text{rank}(\mathfrak{g})$ and $j=1,...,\text{dim(R)}$.