Say I have a finite Weyl group, $W$, and a set of generators $S:= \{s_1,...,s_k\}$ (making $W,S$ a coxeter system) and an automorphism $\theta: W\rightarrow W$ which permutes $S$. I know that the fixed subgroup $W^\theta$ is a Weyl group generated by the longest elements in the subgroups generated by the $\theta$ orbits of the elements of $S$. Suppose I had a longest element $w$, in $W$ with length $m$. And lets say it happens that $w\in W^\theta$ and also lets say that the factorization $w = s_1...s_m$ can be grouped: $$(s_1...s_{k_1})(s_{k_1+1}...s_{k_{2}})...(s_{k_{j-1}+1}...s_{k_j})$$ in such a way that the products in parentheses are generators for $W^\theta$. Is $w$ also a longest element for $W^\theta$ with length $j$ (note that $j$ is the number of parenthetical groups in the centered line)?
If not, is there any general connection between the longest element of the big group and that of the fixed subgroup? Or between their respective lengths?