Maximal length elements of excess zero in finite Coxeter groups

In this paper we prove that for W a finite Coxeter group and C a conjugacy class of W, there is always an element of C of maximal length in C which has excess zero. An element w∈W has excess zero if there exist elements σ,τ∈W such that σ2=τ2=1,w=στ and ℓ(w)=ℓ(σ)+ℓ(τ), ℓ being the length function on W.