In a talk at the Banff International Research Station in 2015, Asher Auel asked questions about genus one curves in Severi-Brauer varieties
S
B
(
A
)
SB(A)
. More specifically, he asked about the smooth cubic curves in Severi-Brauer surfaces, that is, in
S
B
(
D
)
SB(D)
where
D
/
F
D/F
is a degree three division algebra. Even more specifically, he asked about the Jacobian,
E
E
, of these curves. In this paper we give a version of an answer to both these questions, describing the surprising connection between these curves and properties of the algebra
A
A
. Let
F
F
contain
ρ
\rho
, a primitive third root of one. Since
D
/
F
D/F
is cyclic, it is generated over
F
F
by
x
,
y
x,y
such that
x
y
=
ρ
y
x
xy = \rho {yx}
and we call
x
,
y
x,y
a skew commuting pairs. The connection mentioned above is between the Galois structure of the three torsion points
E
[
3
]
E[3]
and the Galois structure of skew commuting pairs in extensions
D
⊗
F
K
D \otimes _F K
. Given a description of which
E
E
arise, we then describe, via Galois cohomology, which
C
C
arise.