  We analyse the complexity of constructing involution centralisers in unitary
groups over fields of odd order. In particular, we prove logarithmic bounds on
the number of random elements required to generate a subgroup of the
centraliser of a strong involution that contains the last term of its derived
series. We use this to strengthen previous bounds on the complexity of
recognition algorithms for unitary groups in odd characteristic. Our approach
generalises and extends two previous papers by the second author and
collaborators on strong involutions and regular semisimple elements of linear
groups.
