  Let $W$ be a finite Coxeter group. It is well-known that the number of
involutions in $W$ is equal to the sum of the degrees of the irreducible
characters of $W$. Following a suggestion of Lusztig, we show that this
equality is compatible with the decomposition of $W$ into Kazhdan--Lusztig
cells. The proof uses a generalisation of the Frobenius--Schur indicator to
symmetric algebras, which may be of independent interest.
