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We continue our study of Cartan schemes and their Weyl group-oids and obtain a complete list of all connected simply connected Cartan schemes of rank three for which the real roots form a finite irreducible root system. We achieve this result by providing an algorithm which determines all the root systems and eventually terminates: Up to equivalence there are exactly 55 such Cartan schemes, and the number of corresponding real roots varies between and . We identify those Weyl groupoids which appear in the classification of Nichols algebras of diagonal type. References