  The purpose of this paper is to study reducibility properties in Sasakian
geometry. First we give the Sasaki version of the de Rham Decomposition
Theorem; however, we need a mild technical assumption on the Sasaki
automorphism group which includes the toric case. Next we introduce the concept
of {\it cone reducible} and consider $S^3$ bundles over a smooth projective
algebraic variety where we give a classification result concerning contact
structures admitting the action of a 2-torus of Reeb type. In particular, we
can classify all such Sasakian structures up to contact isotopy on $S^3$
bundles over a Riemann surface of genus greater than zero. Finally, we show
that in the toric case an extremal Sasaki metric on a Sasaki join always
splits.
