Invariant expressions for linear complex mechanisms: Generalization for polar two-step sub-mechanisms

•A new linear sub-mechanism is characterized: the polar two-step sub-mechanism.•This two-step sub-mechanism can be identified within a complex linear mechanism.•Invariant expressions were obtained for the polar two-step sub-mechanism.•These invariant expressions are identical to those obtained from a two-step mechanism.•A mathematical demonstration of this feature is presented. We generalize the results on invariances found for linear two-step mechanism within our approach to joint kinetics. It is demonstrated that similar invariances can be formulated for complex linear mechanisms if within these mechanisms there is a specific sub-mechanism, i.e. a so-called polar two-step sub-mechanism involving A ⇄ B ⇄ C where B can be part of a linear sub-mechanism not otherwise involving the poles A and C.