Let
R
⟨
2
⟩
=
R
⟨
x
,
y
⟩
{R_{\left \langle 2 \right \rangle }} = R\left \langle {x,y} \right \rangle
be the free associative algebra of rank 2, on the free generators
x
x
and
y
y
, over
R
R
(
R
R
a field, a Euclidean domain, etc.). We prove that if
φ
\varphi
is an automorphism of
R
⟨
2
⟩
{R_{\left \langle 2 \right \rangle }}
that keeps
(
x
y
−
y
x
)
(xy - yx)
fixed (up to multiplication by an element of
R
R
), then
φ
\varphi
is tame, i.e. it is a product of elementary automorphisms of
R
⟨
2
⟩
{R_{\left \langle 2 \right \rangle }}
. This follows from a more general result about endomorphisms of
R
⟨
2
⟩
{R_{\left \langle 2 \right \rangle }}
via a theorem due to H. Jung [6] concerning automorphisms of a commutative and associative algebra of rank 2.