We use the invariance of the triangle
T
2
=
{
(
x
,
y
)
∈
R
2
:
0
⩽
x
,
y
,
1
−
x
−
y
}
\mathbf {T}^2=\{(x,y)\in \mathbb {R}^2:\, 0\leqslant x,y,\, 1-x-y\}
under the permutations of
{
x
,
y
,
1
−
x
−
y
}
\{x,y,1-x-y\}
to construct and study two-variable orthogonal polynomial systems with respect to several distinct Sobolev inner products defined on
T
2
\mathbf {T}^2
. These orthogonal polynomials can be constructed from two sequences of univariate orthogonal polynomials. In particular, one of the two univariate sequences of polynomials is orthogonal with respect to a Sobolev inner product and the other is a sequence of classical Jacobi polynomials.