I've shown, quite accidentally, that Weyl group of $F_4$ injects into the Weyl group of $E_6$ as the subgroup of elements normalizing a maximal torus $T^4$ of $F_4$. One might a priori expect other Weyl elements of $E_6$ to normalize this torus, but it isn't the case.
I came to this result through number crunching and realizing that the Weyl group of $E_6$ acts simply transitively on ordered quadruples of mutually orthogonal roots. The same statement doesn't hold for the inclusion $D_4 \to E_6$, for instance, so it doesn't seem like it should have been a priori obvious before I did this calculation.
- Is it actually a priori obvious?
In any event, it must surely be known.
- Where can I find a citation?
Update: I've received some responses on MathOverflow: https://mathoverflow.net/questions/203295/is-this-characterization-of-1-eigenspaces-of-the-weyl-group-of-e-6-known.