Let X1 and X2 be two compact connected strongly pseudoconvex embeddable Cauchy-Riemann
(CR) manifolds of dimensions 2m − 1 and 2n − 1 in Cm+1 and Cn+1, respectively. We introduce the Thom-
Sebastiani sum X = X1⊕X2 which is a new compact connected strongly pseudoconvex embeddable CR manifold
of dimension 2m+2n+1 in Cm+n+2. Thus the set of all codimension 3 strongly pseudoconvex compact connected
CR manifolds in Cn+1 for all n > 2 forms a semigroup. X is said to be an irreducible element in this semigroup
if X cannot be written in the form X1 ⊕X2. It is a natural question to determine when X is an irreducible CR
manifold. We use Kohn-Rossi cohomology groups to give a necessary condition of the above question. Explicitly,
we show that if X = X1 ⊕ X2, then the Kohn-Rossi cohomology of the X is the product of those Kohn-Rossi
cohomology coming from X1 and X2 provided that X2 admits a transversal holomorphic S1-action.