  Our aim is to support the choice of two remarkable connections with torsion
in a 3-Sasakian manifold, proving that, in contrast to the Levi-Civita
connection, the holonomy group in the homogeneous cases reduces to a proper
subgroup of the special orthogonal group, of dimension considerably smaller. We
realize the computations of the holonomies in a unified way, by using as a main
algebraic tool a nonassociative structure, that one of symplectic triple
system.
