  We continue our study of Cartan schemes and their Weyl groupoids. The results
in this paper provide an algorithm to determine connected simply connected
Cartan schemes of rank three, where the real roots form a finite irreducible
root system. The algorithm terminates: Up to equivalence there are exactly 55
such Cartan schemes, and the number of corresponding real roots varies between
6 and 37. We identify those Weyl groupoids which appear in the classification
of Nichols algebras of diagonal type.
