  We obtain an inequality for the kissing number in 16 dimensions. We do this
by generalising a sum-product bound of Solymosi and Wong for quaternions to a
semialgebra in dimension 16. In particular, we obtain the inequality
$$k_{16}\geq \frac{\sum_{x \in
\mathcal{R}}\left|\mathcal{S}_{x}\right|}{\left|\bigcup_{x \in \mathcal{R}}
\mathcal{S}_{x}\right|}-1,$$ where $k_{16}$ is the 16-dimensional kissing
number, and $S_x$ and $\mathcal{R}$ are sets defined below. Along the way we
also obtain a sum-product bound for subsets of the octonions which are closed
under taking inverses, using a similar strategy to that used for the
quaternions. We use the fact that the kissing number in eight dimensions is 240
to achieve the result. Namely, we obtain the bound
$$\operatorname{max}(|\mathcal{A}+\mathcal{A}|,|\mathcal{A}\mathcal{A}|)\geq
\frac{|\mathcal{A}|^{4/3}}{(1928\cdot\lceil
\operatorname{log}|\mathcal{A}|\rceil)^{1/3}},$$ where $\mathcal{A}$ is a
finite set of octonions such that if $x\in\mathcal{A}$, $x^{-1}\in\mathcal{A}$
also.
