Differential algebra for derivations with nontrivial commutation rules

The classical assumption of differential algebra, differential elimination theory and formal integrability theory is that the derivations do commute. This is the standard case arising from systems of partial differential equations written in terms of the derivations w.r.t. the independant variables. We inspect here the case where the derivations satisfy nontrivial commutation rules. Such a situation arises, for instance, when we consider a system of equations on the differential invariants of a Lie group action. We develop the algebraic foundations for such a situation. They lead to algorithms for completion to formal integrability and differential elimination.