I'm trying to do some explicit computations for which I need to know the maximal Levi subgroups of exceptional complex Lie groups $\mathrm{F}_4=\mathrm{F}_4(\mathbb{C})$.
Of course from the Dynkin diagram I can see for example that one maximal sub-root system is of type $B_3$ and another is of type $C_3$. By the general theory, their corresponding Levi subgroups $M$ are connected reductive of type $B_3$ and $C_3$. But the maximal torus in the center $Z(M)$ of $M$ is one dimensional. It's natural to guess that for the type $C_3$ I could get $M=\mathrm{GSp}(6)$, but I could not find a reference for this.
Could any veteran point out some related references for me (about Levi subgroups of exceptional Lie groups)? And I wonder is there a database for $\mathrm{E}_6$, $\mathrm{E}_7$, $\mathrm{E}_8$?
Thanks a lot in advance!