  Feit and Fine derived a generating function for the number of ordered pairs
of commuting n by n matrices over the finite field F_q. This has been reproved
and studied by Bryan and Morrison from the viewpoint of motivic
Donaldson-Thomas theory. In this note we give a new proof of the Feit-Fine
result, and generalize it to the Lie algebra of finite unitary groups and to
the Lie algebra of odd characteristic finite symplectic groups. We extract some
asymptotic information from these generating functions. Finally, we derive
generating functions for the number of commuting nilpotent elements for the Lie
algebras of the finite general linear and unitary groups, and of odd
characteristic symplectic groups.
