The automorphism group of a completely positive cone and its Lie algebra

Given a closed cone C in Rn, we consider the corresponding completely positive (convex) cone K generated by {uuT:u∈C} in Sn. Under certain conditions on C, we describe the automorphism group of K and its corresponding Lie algebra in terms of those of C∪-C and/or C. In particular, we show that when C is a (closed convex) proper cone, the automorphism groups of C and K are isomorphic and their corresponding Lie algebras are isomorphic.