manuscripta math. © Springer-Verlag 2009 Ágota Figula · Karl Strambach Subloop incompatible Bol loops Received: 10 December 2008 / Revised: 10 April 2009 Abstract. We give a necessary modification of Proposition 1.18 in Nagy and Strambach (Loops in Group Theory and Lie Theory. de Gruyter Expositions in Mathematics Berlin, New York, 2002) and close the gap in the classification of differentiable Bol loops given in Figula (Manuscrp Math 121:367–385, 2006). Moreover, using the factorization of Lie groups we determine the simple differentiable proper Bol loops L having the direct product G 1 × G 2 of two groups with simple Lie algebras as the group topologically generated by their left translations such that the stabilizer of the identity element of L is the direct product H 1 × H 2 with H i < G i . Also if G 1 = G 2 = G is a simple permutation group containing a sharply transitive subgroup A, then an analogous construction yields a simple proper Bol loop. If A is cyclic and G is finite and primitive, then all such loops are classified. 1. Introduction In [12] the loops L are consistently considered as sharply transitive sections σ : G/ H → G, where G is the group generated by the left translations of L and H is the stabilizer of the identity element e of L in G. This point of view is applied there for a classification of differentiable loops of low dimension. Using the methods of [12] in [3] a classification of differentiable Bol loops having an at most nine-dimensional semi-simple Lie group as the group topologically generated by their left translations is given. A useful tool proving this classification was Proposition 1.18 in [12]: If the group G generated by the left translations of a loop L is the direct product G = G 1 × G 2 and for the stabilizer H of e ∈ L in G one has H = H 1 × H 2 with H i < G i , then L is a product of two loops L 1 and L 2 . But the further claim of this proposition that the loop L i , i = 1, 2, is isomorphic to a loop having G i as the group generated by its left translations and H i as the stabilizer of the identity needs a modification (see Proposition 1). Namely, there are loops L = L 1 L 2 , which we call subloop incompatible loops such that at least one of the subgroups generated by the left translations of L i , i = 1, 2, is a proper subgroup of G i . Á. Figula (B ): Institute of Mathematics, University of Debrecen, P.O. Box 12, 4010 Debrecen, Hungary. e-mail: figula@math.klte.hu K. Strambach: Department Mathematik, Universität Erlangen-Nürnberg, Bismarkstr. 1 1/2, 91054 Erlangen, Germany. e-mail: strambach@mi.uni-erlangen.de Mathematics Subject Classification (2000): 53C30, 53C35, 20N05, 22E15 DOI: 10.1007/s00229-009-0279-y