  We consider a manifestly Lorentz invariant form $\mathbb L$ of the
biquaternion algebra and its generalization to the case of curved manifold. The
conditions of $\mathbb L$-differentiability of $\mathbb L$-functions are
formulated and considered as the primary equations for fundamental fields
modeled with such functions. The exact form of the effective affine connection
induced by $\mathbb L$-differentiability equations is obtained for the flat and
curved cases. In the flat case, the integrability conditions of the latter lead
to the self-duality of the corresponding curvature, thus ensuring that the
source-free Maxwell and $SL(2,\mathbb C)$ Yang-Mills equations hold on the
solutions of the $\mathbb L$-differentiability equations
