  Full details are given for the definition and construction of the wreath
product of two arbitrary Lie algebras, in the hope that it can lead to the
definition of a suitable Lie group to be the wreath product of two given Lie
groups. In the process, quite a few new notions are needed, and introduced.
Such are, for example : Formal series with variables in a vector space and
coefficients in some other vector space. Derivation of a formal series relative
to another formal series. The Lie algebra of a vector space. Formal actions of
Lie algebras over vector spaces. The basic formal action of a Lie algebra over
itself (as a formal version of the analytic aspect of the infinitesimal
operation law of a Lie groupuscule). More generally, the wreath product of two
Lie algebras is defined, relative to a formal action of the second onto an
arbitrary vector space. Main features are : A description of the triangular
actions of wreath products over product vector spaces, and a Kaloujnine-Krasner
type theorem : In essence, it says that all Lie extensions of a given Lie
algebra by another Lie algebra are, indeed, subalgebras of their wreath
product.
