On Splitting of the Normalizer of a Maximal Torus in E6(q)

Let G be a finite group of Lie type E6 over Fq (adjoint or simply connected) and W be the Weyl group of G. We describe maximal tori T such that T has a complement in its algebraic normalizer N(G, T). It is well known that for each maximal torus T of G there exists an element w ∈ W such that N(G, T )/T ■ CW(w). When T does not have a complement isomorphic to CW(w), we show that w has a lift in N(G, T) of the same order.