  If a Lie algebra structure g on a vector space is the sum of a family of
mutually compatible Lie algebra structures g_i's, we say that g is simply
assembled from the g_i's. Repeating this procedure with a number of Lie
algebras, themselves simply assembled from the g_i's, one obtains a Lie algebra
assembled in two steps from the g_i's, and so on. We describe the process of
modular disassembling of a Lie algebra into a unimodular and a non-unimodular
part. We then study two inverse questions: which Lie algebras can be assembled
from a given family of Lie algebras, and from which Lie algebras can a given
Lie algebra be assembled? We develop some basic assembling and disassembling
techniques that constitute the elements of a new approach to the general theory
of Lie algebras. The main result of our theory is that any finite-dimensional
Lie algebra over an algebraically closed field of characteristic zero or over R
can be assembled in a finite number of steps from two elementary constituents,
which we call dyons and triadons. Up to an abelian summand, a dyon is a Lie
algebra structure isomorphic to the non-abelian 2-dimensional Lie algebra,
while a triadon is isomorphic to the 3-dimensional Heisenberg Lie algebra. As
an example, we describe constructions of classical Lie algebras from triadons.
