  Tensoring two on-shell super Yang-Mills multiplets in dimensions $D\leq 10$
yields an on-shell supergravity multiplet, possibly with additional matter
multiplets. Associating a (direct sum of) division algebra(s) $\mathbb{D}$ with
each dimension $3\leq D\leq 10$ we obtain formulae for the algebras
$\mathfrak{g}$ and $\mathfrak{h}$ of the U-duality group $G$ and its maximal
compact subgroup $H$, respectively, in terms of the internal global symmetry
algebras of each super Yang-Mills theory. We extend our analysis to include
supergravities coupled to an arbitrary number of matter multiplets by allowing
for non-supersymmetric multiplets in the tensor product.
