  Let $\L_m$ be the scheme of the laws defined by the identities of Jacobi on
$\K^m$. The local studies of an algebraic Lie algebra $\g=\mathrm{R}\ltimes\n$
in $\L_m$ and its nilpotent part $\n$ in the scheme $\L_n^{\mathrm{R}}$ of
$\mathrm{R}$-invariant Lie algebras on $\K^n$ are linked. This comparison is
made by means of slices, which are transversal subschemes to the orbits of $\g$
and $\n$ under the classical groups acting on $\L_m$ and $\L_n^{\mathrm{R}}$
respectively. We prove a reduction theorem saying that, under certain
conditions on $\g$, the local rings of the slices at $\g$ and $\n$ are
isomorphic. In particular, $\g$ is rigid if and only if is $\n$. In the
formalism developed at beginning of this paper, a deformation of $\g$ with base
a local ring $\A$ is a local morphism from the local ring of $\L_m$ at $\g$ to
$\A$. So the study of deformations for a large class of Lie algebras $\g$ in
$\L_m$ is equivalent to that of $\n$ in $\L_n^{\mathrm{R}}$ "modulo" the
actions of groups, which is a more simple problem. The laws of
$\L_n^{\mathrm{R}}$ are nilpotent with the choice of $\mathrm{R}$ and then we
can construct these laws by central extensions. This corresponds to an
induction on the schemes themselves
$\L_n^{\mathrm{R}}\to\L_{n+1}^{\mathrm{R}}$. We restrict this study to a torus
$\mathrm{R}=\mathrm{T}$ for certain slices. This leads to a concept of
continuous families with the possibility to have nilpotent parameters $t$ (the
schemes are generally not reduced). This gives an alternative formalism for the
problem of obstructions classes in the theory of formal deformations of
M.Gerstenhaber. Examples are given with $t^2=0$ ($t\neq 0$) and $t^5=0$
($t^4\neq 0$).
