The automorphism group of a composition of quadratic forms

Let $U \times X \to X$ be a (bilinear) composition $(u,\,x) \mapsto ux$ of two quadratic spaces $U$ and $X$ over a field $F$ of characteristic $\ne 2$ and assume there is a vector in $U$ which induces the identity map on $X$ via this composition. Define $G$ to be the subgroup of $O(U) \times O(X)$ consisting of those pairs $(\phi ,\,\psi )$ satisfying $\phi (u)\psi (x) = \psi (ux)$ identically and define ${G_X}$ to be the projection of $G$ on $O(X)$. The group $G$ is investigated and in particular it is shown that its connected component, as an algebraic group, is isogenous to a product of two or three classical groups and so is reductive. Necessary and sufficient conditions are given for ${G_X}$ to be transitive on the unit sphere of $X$ when $U$ and $X$ are Euclidean spaces.