  The so called Gell-Mann or decontraction formula is proposed as an algebraic
expression inverse to the Inonu-Wigner Lie algebra contraction. It is tailored
to express the Lie algebra elements in terms of the corresponding contracted
ones. In the case of sl(n,R) and su(n) algebras, contracted w.r.t. so(n)
subalgebras, this formula is generally not valid, and applies only in the cases
of some algebra representations. A generalization of the Gell-Mann formula for
sl(n,R) and su(n) algebras, that is valid for all tensorial, spinorial,
(non)unitary representations, is obtained in a group manifold framework of the
SO(n) and/or Spin(n) group. The generalized formula is simple, concise and of
ample application potentiality. The matrix elements of the SL(n,R)/Spin(n),
i.e. SU(n)/SO(n), generators are determined, by making use of the generalized
formula, in a closed form for all irreducible representations.
