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I think I am missing something, because for me, every normal ordered term containing one commutator (like both terms in the right-hand side of the original equation) should be equal to zero.
Actually, I did not demonstrate the last equation I wrote. I have just written the right-hand side of the original equation in an other way. Why the right-hand side $:[\phi_1,\phi_2\phi_3]:$ is non zero ?