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I found the answer myself. Instead of using AAA you should use the SR (Symbolic Ring). An example:

sage: q = var('q')
sage: F.<a,b,c,d> = FreeAlgebra(SR, 4)
sage: (1+q)**(1/2)*b*a
(sqrt(q+1))*b*a

This SR ring is still a bit of a mystery to me, but it seems to work.

I found the answer myself. Instead of using AAA AA you should use the SR (Symbolic Ring). An example:

sage: q = var('q')
sage: F.<a,b,c,d> = FreeAlgebra(SR, 4)
sage: (1+q)**(1/2)*b*a
(sqrt(q+1))*b*a

This SR ring is still a bit of a mystery to me, but it seems to work.