  The concept of spin-base invariance is extended to arbitrary integer
dimension $d \geq 2$. Explicit formulas for the spin connection as a function
of the Dirac matrices are found. We disclose the hidden spin-base invariance of
the vielbein formalism and give a detailed motivation for this symmetry from
first principles. The common Lorentz symmetric gauge for the vielbein is
constructed for the Dirac matrices, even for metrics which are not linearly
connected. Under certain criteria, it constitutes the simplest possible gauge,
demonstrating why this gauge is so useful.
