  We prove some formulas relating Cauchy-Riemann operators defined on
hypercomplex subspaces of an alternative *-algebra to a differential operator
associated with the concept of slice-regularity and to the spherical Dirac
operator. These results in particular allow to introduce a definition of
locally slice-regular function and open the path for local slice analysis.
Since Cauchy-Riemann operators factor the corresponding Laplacian operators,
the proven formulas let us also obtain several results about the harmonicity
and polyharmonicity properties of slice-regular functions.
