Like others, I'm very sorry to learn of Jerry Marsden's passing. He kindly helped all who asked (including me) despite a very busy life and recent illness. It seems fitting to mention this before citing him below.
According to Marsden/Hughes (Mathematical Foundations of Elasticity), stretch & shear (from homogeneous elastic deformation) are determined soley by U in the polar decomposition of the deformation gradient F=RU where R = rot'n & U = right stretch.
Using SVD decomposition (same as eigenvalue decomposition as U positive definite), U can be decomposed into U = PEP^T where P = matrix of eigenvectors of U (as columns) & E = diagonal matrix of eigenvalues of U (i.e. principal stretches). The columns of P are orthogonal as are P & P^T (i.e. PP^T=I) so P, P^T are rot'ns. The PE portion of PEP^T produces combined stretching / shearing.
Based on this, I'm wondering if the free configuration manifold for homogeneous deformation can be expressed as: SO(3) X DIAG(3) X SO(3) where DIAG(3) = diag. 3 X 3 "pure stretch" matrices with non-zero diag. entries. If so, I think DIAG(3) is a subgroup of GL(3, R) so a Lie group with its Lie algebra being 3x3 diag. matrices with real diag. entries.
Any thoughts/opinions about this would be very welcome.
Thanks, John.