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It has been shown that for several nilpotent Gelfand pairs (N,K) (i.e., withN a nilpotent Lie group, K a compact group of automorphisms of N and thealgebra L^1(N)^K commutative) the spherical transform establishes a 1-to-1correspondence between the space S(N)^K of K-invariant Schwartz functions on Nand the space S({\Sigma}) of functions on the Gelfand spectrum {\Sigma} ofL^1(N)^K which extend to Schwartz functions on Rd, once {\Sigma} is suitablyembedded in Rd. We call this property (S). We present here a generalbootstrapping method which allows to establish property (S) to new nilpotentpairs (N,K), once the same property is known for a class of quotient pairs of(N,K) and a K-invariant form of Hadamard formula holds on N. We finally showhow our method can be recursively applied to prove property (S) for asignificant class of nilpotent Gelfand pairs.

Nilpotent gelfand pairs and Schwartz extensions of spherical transforms via quotient pairs