On soluble skew linear groups over finite-dimensional division algebras

Let D be a division ring of finite degree d and let n be a positive integer. If G is any soluble subgroup of image, we prove that G has derived length at most 9+log2d+(11/3)log2n and that G has a unipotent-by-abelian (abelian if G is completely reducible) normal subgroup of finite index dividing b(n).d2n, where b(n) is an integer-valued function of n only. Actually, we derive bounds rather better than those quoted above, but rather more involved to state.