  The normalizer $N_G(H_G)$ of a maximal torus $H_G$ in a semisimple complex
Lie group $G$ does not in general allow a presentation as a semidirect product
of $H_G$ and the corresponding Weyl group $W_G$. Meanwhile, splitting holds for
classical groups corresponding to the root systems $A_\ell$, $B_\ell$,
$D_\ell$. For the remaining classical groups corresponding to the root systems
$C_\ell$ there still exists an embedding of the Tits extension of $W_G$ into
normalizer $N_G(H_G)$. We provide explicit unified construction of the lifts of
the Weyl groups into normalizers of maximal tori for classical Lie groups
corresponding to the root systems $A_\ell$, $B_\ell$, $D_\ell$ using embeddings
into general linear Lie groups. For symplectic series of classical Lie groups
we provide an explanation of impossibility of embedding of the Weyl group into
the symplectic group. The explicit formula for adjoint action of the lifts of
the Weyl groups on $\mathfrak{g}={\rm
  Lie}(G)$ are given. Finally some examples of the groups closely associated
with classical Lie groups are considered.
