In general it is not possible to embed the Weyl group W in the group G: already you can see this for SL2(mathbbC), where the Weyl group has order 2: if the torus fixes the lines spanned by e1 and e2 respectively, you want to pick the linear map taking e1 to e2 and e2 to e1, but this has determinant −1. A lift of W to N(T) must be an element of order 4 not 2, say e1mapsto−e2 and e2mapstoe1.
In fact, Tits has shown that this is essentially the only obstruction: the Weyl group can always be lifted to a group tildeW inside G which is an extension of W by an elementary abelian 2-group of order 2l where l is the number of simple roots. If I recall correctly, this lift is then unique up to conjugation.
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