  We study symmetric groups and left braces satisfying special conditions, or
identities. We are particularly interested in the impact of conditions like
$\textbf{Raut}$ and $\textbf{lri}$ on the properties of the symmetric group and
its associated brace. We show that the symmetric group $G=G(X,r)$ associated to
a nontrivial solution $(X,r)$ has multipermutation level $2$ if and only if $G$
satisfies $\textbf{lri}$. In the special case of a two-sided brace we express
each of the conditions $\textbf{lri}$ and $\textbf{Raut}$ as identities on the
associated radical ring $G_*$. We apply these to construct examples of
two-sided braces satisfying some prescribed conditions. In particular we
construct a finite two-sided brace with condition $\textbf{Raut}$ which does
not satisfy $\textbf{lri}$. (It is known that condition $\textbf{lri}$ implies
$\textbf{Raut}$). We show that a finitely generated two-sided brace which
satisfies \textbf{lri} has a finite multipermutation level which is bounded by
the number of its generators.
