  We develop a mathematical concept towards gauge field theories based upon a
Hilbert space endowed with a representation of a skew-adjoint Lie algebra and
an action of a generalized Dirac operator. This concept shares common features
with the non-commutative geometry a la Connes/Lott, differs from that, however,
by the implementation of skew-adjoint Lie algebras instead of unital
associative *-algebras. We present the physical motivation for our approach and
sketch its mathematical strategy. Moreover, we comment on the application of
our method to the standard model and the flipped SU(5) x U(1)-grand unification
model.
