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