  Recently, the geodesibility of planar vector fields, which are algebrizable
(differentiable in the sense of Lorch for some associative and commutative
unital algebra), has been established. In this paper, we consider algebrizable
three-dimensional vector fields, for which we give rectifications and
Riemannian metrics under which they are geodesible. Furthermore, for each of
these vector fields $F$ we give two first integrals $h_1$ and $h_2$ such that
the integral curves of $F$ are locally defined by the intersections of the
level surfaces of $h_1$ and $h_2$.
