  This elementary discussion generalizes a Weyl geometry to allow quaternion
valued gauge transformations and classical Yang-Mills geometric fields. This
development will assume that the symmetric metric tensor is real in some gauge,
and will develop the left and right handed approaches to quaternionic gauge
transformations. Quaternionic gauge transformations are shown actually to
require the shifting of some of Weyl's nonmetricity into torsion to define a
properly transforming gauge field full curvature tensor, which is constructed
as an asymmetric sum of left and right handed forms. Natural, gauge invariant,
dimensionless variables are defined suitable for physics, and for use as a
general formalism to describe these geometries, including General Relativity,
in rather general circumstances. The geometry "self measures" these variables.
Weyl's original action principle provides an example of an action rephrased in
these gauge invariant variables, along with some unexpected possible insights
on mechanics promoted by such a formulation of that action. Those include the
torsion tensor and nonmetricity being constructed from mechanical
energy-momentum. The Weyl form of action is then generalized to a quaternionic
gauge field. The insights on mechanics now include spin 1/2 Dirac free fields.
For physically reasonable choices of free parameters, the dimensionless Ricci
tensor becomes nonnegligible in particle physics at distances much greater than
the Planck length, along with limited general relativistic effects.
