  In 2005, Nualart and Peccati showed that, surprisingly, the convergence in
distribution of a normalized sequence of multiple Wiener-It\^o integrals
towards a standard Gaussian law is equivalent to convergence of just the fourth
moment to 3. Recently, this result has been extended to a sequence of multiple
Wigner integrals, in the context of free Brownian motion. The goal of the
present paper is to offer an elementary, unifying proof of these two results.
The only advanced, needed tool is the product formula for multiple integrals.
Apart from this formula, the rest of the proof only relies on soft
combinatorial arguments.
