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With his Clifford algebra of differential forms, Kaehler's algebra addressesthe overlooked manifestation of symmetry in the solutions of exterior systems.In this algebra, solutions with a given symmetry are members of left idealsgenerated by corresponding idempotents. These combine with phase factors totake care of all the dependence of the solution on $x^{i}$ and $% dx^{i}$ foreach $\partial /\partial x^{i}=0$ symmetry. The maximum number of idempotents(each for a one-parameter group) that can, therefore, go into a solution is thedimensionality $n$ of the space. This number is further limited bynon-commutativity of idempotents.  We consider the tensor product of Kaehler algebra with tangent Cliffordalgebra, i.e. of valuedness. It has a commutative subalgebra of "mirrorelements", i.e. where the valuedness is dual to the differential form (like in$dx\mathbf{i}$, but not in $dx\mathbf{j}$). It removes the aforementionedlimitation in the number of idempotent factors that can simultaneously go intoa solution.  In the Kaehler calculus, which is based on the Kaehler algebra, total angularmomentum is linear in its components, even it is not valued in tangent algebra.In particular, it is not a vector operator.  We carry Kaehler's treatment over to the aforementioned commutative algebra.Non-commutativity thus ceases to be a limiting factor in the formation ofproducts of idempotents representing each a one-parameter symmetry. Theinterplay of factors that emerges in the expanded set of such products isinimical to operator theory. For practical reasons, we stop at productsinvolving three one-parameter idempotents ---henceforth call ternary--- eventhough the dimensionality of the all important spacetime manifold allows forfour of them.

Kaehler Algebra: Idempotents for Solutions with Symmetry in Metric Spaces