  It is proved that the Nagata automorphism (Nagata coordinates, respectively)
of the polynomial algebra $F[x,y,z]$ over a field $F$ cannot be lifted to a
$z$-automorphism ($z$-coordinate, respectively) of the free associative algebra
$K<x,y,z>$. The proof is based on the following two new results which have
their own interests: degree estimate of ${Q*_FF<x_1,...,x_n>}$ and tameness of
the automorphism group ${\text{Aut}_Q(Q*_FF<x,y>)}$.
