  We study the action of the inertia operator on the motivic Hall algebra, and
prove that it is diagonalizable. This leads to a filtration of the Hall
algebra, whose associated graded algebra is commutative. In particular, the
degree 1 subspace forms a Lie algebra, which we call the Lie algebra of
virtually indecomposable elements, following Joyce. We prove that the integral
of virtually indecomposable elements admits an Euler characteristic
specialization. In order to take advantage of the fact that our inertia groups
are unit groups in algebras, we introduce the notion of algebroid.
