In [4] this result was proved by using Theorem 3 of [I]. We now give a new proof, independent of the results of [I]. The present proof has the advantage of being constructive, i.e. we actually show how to construct a certain subgroup of 7r which is required in the proof of Theorem A. The proofs of Wolf [5] are different from both [4] and the present proofs. The main results of this paper were obtained by J. A. Wolf and the author independently, see [5 ]. If a group G of automorphisms of an n-dimensional real vector space leaves complementary subspaces A and B invariant, then G restricted to A and B shall be denoted by p(G) and q(G)respectively. We shall write G = p (G) +q (G).