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The existence of A$_\rho$-loops, A$_\lambda$-loops and A$_\mu$-loops that areneither extra loops nor CC-loops such that any two of their inner mappings$R(x,y),L(x,y)$ and $T(x)$ commute while the other one is of order 2 is shown.

On the existence of A-loops with some commutative inner mappings and others of order 2