Let A be a finite-dimensional algebra over the rational numbers. The Wedderburn-Maltsev theorem tates that A decomposes as a sum of a semisimple subring and the Jacobson radical of A. In this aper the authors study the consequences of the Wedderburn-Maltsev theorem for unit groups of classical orders in such an algebra. The authors call the subgroup of a group consisting of all elements with finitely many conjugates the finite conjugacy centre. A priori this need not be a central subgroup.