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Let \mathcal{W} be the set of strongly real elements of W, a Coxeter group. Then for $w \in \mathcal{W}$,\ud $e(w)$, the excess of w, is defined by \ud $e(w) = \min min \{l(x)+l(y) - l(w)| w = xy; x^2 = y^2 =1}$. When $W$ is finite we may also define E(w), the reflection excess of $w$. The main result established here is that if $W$ is finite and $X$ is a $W$-conjugacy class, then there\ud exists $w \in X$ such that $w$ has minimal length in $X$ and $e(w) = 0 = E(w)$

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