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Under mild conditions we show that the affinity dimension of a planarself-affine set is equal to the supremum of the Lyapunov dimensions ofself-affine measures supported on self-affine proper subsets of the originalset. These self-affine subsets may be chosen so as to have stronger separationproperties and in such a way that the linear parts of their affinities arepositive matrices. Combining this result with some recent breakthroughs in thestudy of self-affine measures and their associated Furstenberg measures, weobtain new criteria under which the Hausdorff dimension of a self-affine setequals its affinity dimension. For example, applying recent results ofB\'{a}r\'{a}ny, Hochman-Solomyak and Rapaport, we provide new explicit examplesof self-affine sets whose Hausdorff dimension equals its affinity dimension,and for which the linear parts do not satisfy any domination assumptions.

On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems