  We prove that the finite exceptional groups $F_4(q)$, $E_7(q)_{\mathrm{ad}}$,
and $E_8(q)$ have no irreducible complex characters with Frobenius-Schur
indicator $-1$, and we list exactly which irreducible characters of these
groups are not real-valued. We also give an exact list of complex irreducible
characters of the Ree groups ${^2 F_4}(q^2)$ which are not real-valued, and we
show the only character of this group which has Frobenius-Schur indicator $-1$
is the cuspidal unipotent character $\chi_{21}$ found by M. Geck.
