Let
R
R
be a commutative domain with 1.
R
⟨
x
,
y
⟩
R\langle x,y\rangle
stands for the free associative algebra of rank 2 over
R
;
R
[
x
~
,
y
~
]
R;R[\tilde x,\tilde y]
is the polynomial algebra over
R
R
in the commuting indeterminates
x
~
\tilde x
and
y
~
\tilde y
. We prove that the map
Ab
:
Aut
(
R
⟨
x
,
y
⟩
)
→
Aut
(
R
[
x
~
,
y
~
]
)
\text {Ab}: \operatorname {Aut} (R\langle x,y\rangle ) \to \operatorname {Aut} (R[\tilde x,\tilde y])
induced by the abelianization functor is a monomorphism. As a corollary to this statement and a theorem of Jung [5], Nagata [7] and van der Kulk [8]* that describes the automorphisms of
F
[
x
~
,
y
~
]
F[\tilde x,\tilde y]
(
F
F
a field) we are able to conclude that every automorphism of
F
⟨
x
,
y
⟩
F\langle x,y\rangle
is tame (i.e. a product of elementary automorphisms).