  In this paper we determine the at least $4$-dimensional affine reductive
homogeneous manifolds for an at most $9$-dimensional simple Lie group or an at
most $6$-dimensional semi-simple Lie group. Those reductive spaces among them
which admit a sharply transitive differentiable section yield local almost
differentiable left A-loops. Using this we classify all global almost
differentiable left A-loops $L$ having either a $6$-dimensional semi-simple Lie
group or the group $SL_3(\mathbb R)$ as the group topologically generated by
their left translations. Moreover, we determine all at most $5$-dimensional
left A-loops $L$ with $PSU_3(\mathbb C,1)$ as the group topologically generated
by their left translations.
