  We prove that for a number field $F$, the distribution of the points of a set
$\Sigma \subset \mathbb{A}_F^n$ with a purely exponential parametrization, for
example a set of matrices boundedly generated by semi-simple (diagonalizable)
elements, is of at most logarithmic size when ordered by height. As a
consequence, one obtains that a linear group $\Gamma\subset \mathrm{GL}_n(K)$
over a field $K$ of characteristic zero admits a purely exponential
parametrization if and only if it is finitely generated and the connected
component of its Zariski closure is a torus. Our results are obtained via a key
inequality about the heights of minimal $m$-tuples for purely exponential
parametrizations. One main ingredient of our proof is Evertse's strengthening
of the $S$-Unit Equation Theorem.
