  Let $(W,S)$ be a Coxeter system and $\ast$ be an automorphism of $W$ with
order $\leq 2$ such that $s^{\ast}\in S$ for any $s\in S$. Let $I_{\ast}$ be
the set of twisted involutions relative to $\ast$ in $W$. In this paper we
consider the case when $\ast=\text{id}$ and study the braid
$I_\ast$-transformations between the reduced $I_\ast$-expressions of
involutions. If $W$ is the Weyl group of type $B_n$ or $D_n$, we explicitly
describe a finite set of basic braid $I_\ast$-transformations for all $n$
simultaneously, and show that any two reduced $I_\ast$-expressions for a given
involution can be transformed into each other through a series of basic braid
$I_\ast$-transformations. In both cases, these basic braid
$I_\ast$-transformations consist of the usual basic braid transformations plus
some natural "right end transformations" and plus exactly one extra
transformation. The main result generalizes our previous work for the Weyl
group of type $A_{n}$.
