Almost incompressible hyperelastic theory

In Abaqus, when the material response is almost incompressible, Abaqus/Standard offers a “mixed” formulation.

I think this formulation is the three-field variational principle originally proposed by SlMO known as the mixed Jacobian-pressure formulation which emanates from a three-field variational principle of Hu and WASHIZU. The three independent field variables are displacement U, pressure P and volume dilatation J, respectively. But, In the derivation process in Abaqus, variations of Lagrange multiplier contains

which means P and J are not independently.

I want to know what exactly the variational principle Abaqus use and what are the independent field variables in the formulation.