  Continual Lie algebras are infinite dimensional generalizations of ordinary
Lie with continuous set of roots. In this paper we establish a general relation
between chain complexes and continual Lie algebras. First, by defining and
appropriate product among a chain complex spaces we endow it with the structure
of an exterior differential graded algebra $\mathcal C$. The natural
orthogonality condition with respect to a product bring about to $\mathcal C$
the structure of a graded algebra with differential relations and compatibility
conditions for elements of chain complex spaces. By considering the union of
all relations for independent elements in the tree graph structure of the
system, we prove the main result of this paper: a chain complex brings about
the structure of a continual Lie algebra with the root space determined by
parameters for the complex. This provides a new source of examples of continual
Lie algebras. Finally, as an example, we consider the case of \v{C}ech-de Rham
complex associated to a foliation on a smooth manifold. In a particular case of
this chain complex, we derive explicitly the commutation relations for
corresponding continual Lie algebra.
