Htykuut

In this question it was shown that the simply connected compact Lie group of type $E_6$ the subgroup $Gamma$ of maximal rank of type $A_2^3$ given by Borel-de Siebenthal theory (see the question) is isomorphic to $SU(3)^3/H$ where $H$ is a finite group of order $3$. This was proved by computing the order of the center of $Gamma$ (it turned out to be $9$).

My question is the following: which subgroup of $SU(3)^3$ is $H$? And what is the center of $E_6$ (it is isomorphic to $mu_3$) inside $SU(3)/H$?

The most canonical choice for $H$ seems to be ${(a,b,c)inmu_3^3:abc=1}$, and the most canonical choice for the center seems ${(a,a,a):ain mu_3}$, but this are not compatible, because I am not killing anything when I go to the adjoint form of $E_6$.

But maybe one of the three copies of $SU(3)$ -the one generated by the highest root and a simple root- is distinguished, so that the situation is not really symmetric... I feel terrible because I have no idea how to check any of these things...