  We develop a new method in order to classify the Bianchi I spacetimes which
admit conformal Killing vectors (CKV). The method is based on two propositions
which relate the CKVs of 1+(n-1) decomposable Riemannian spaces with the CKVs
of the (n-1) subspace and show that if 1+(n-1) space is conformally flat then
the (n-1) spacetime is maximally symmetric. The method is used to study the
conformal algebra of the Kasner spacetime and other less known Bianchi type I
matter solutions of General Relativity.
