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Given a complex connected reductive Lie group  G  with a maximal torus    $$H\subset G$$  H ⊂ G , Tits defined an extension    $$W_G^{\mathrm{T}}$$  W G T of the corresponding Weyl group    $$W_G$$  W G . The extended group is supplied with an embedding into the normalizer    $$N_G(H)$$  N G ( H ) such that    $$W_G^{\mathrm{T}}$$  W G T together with  H  generate    $$N_G(H)$$  N G ( H ) . In this paper we propose an interpretation of the Tits classical construction in terms of the maximal split real form    $$G(\mathbb {R})\subset G$$  G ( R ) ⊂ G , which leads to a simple topological description of    $$W^{\mathrm{T}}_G$$  W G T . We also consider a variation of the Tits construction associated with compact real form  U  of  G . In this case we define an extension    $$W_G^U$$  W G U of the Weyl group    $$W_G$$  W G , naturally embedded into the group extension    $$\widetilde{U}:=U\,{\rtimes }\, \Gamma $$  U ~ : = U ⋊ Γ of the compact real form  U  by the Galois group    $$\Gamma ={\mathrm{Gal}}(\mathbb {C}/\mathbb {R})$$  Γ = Gal ( C / R ) . Generators of    $$W^U_G$$  W G U are squared to identity as in the Weyl group    $$W_G$$  W G . However, the non-trivial action of    $$\Gamma $$  Γ by outer automorphisms requires    $$W^U_G$$  W G U to be a non-trivial extension of    $$W_G$$  W G . This gives a specific presentation of the maximal torus normalizer of the group extension    $${\widetilde{U}}$$  U ~ . Finally, we describe explicitly the adjoint action of    $$W_G^{\mathrm{T}}$$  W G T and    $$W^U_G$$  W G U on the Lie algebra of  G .

Normalizers of maximal tori and real forms of Lie groups