Assembling Lie Algebras from Lieons

If a Lie algebra structures $\gG$ on a vector space is the sum of a family of mutually compatible Lie algebra structures $\gG_i$, we say that $\gG$ is \emph{simply assembled} from $\gG_s$'s. By repeating this procedure several times one gets a family of Lie algebras \emph{assembled} from $\gG_s$'s. The central result of this paper is that any finite dimensional Lie algebra over $\R$ or $\C$ can be assembled from two constituents, called $\between$- and $\pitchfork$-\emph{lieons}. A lieon is the direct sum of an abelian Lie algebra with a 2-dmensional nonabelian Lie algebra or with the 3-dimensional Heisenberg algebra. Some techniques of disassembling Lie algebras are introduced and various results concerning assembling-disassembling procedures are obtained. In particular, it is shown how classical Lie algebras are assembled from lieons and is obtained the complete list of Lie algebras, which can be simply assembled from lieons.