  We classify pairs of conjugacy classes in almost simple algebraic groups
whose product consists of finitely many classes. This leads to several
interesting families of examples which are related to a generalization of the
Baer--Suzuki theorem for finite groups. We also answer a question of Pavel
Shumyatsky on commutators of pairs of conjugacy classes in simple algebraic
groups. It turns out that the resulting examples are exactly those for which
the product also consists of only finitely many classes.
