For a classical group
G
G
of type
D
n
\mathsf {D}_n
over a field
k
k
of characteristic different from
2
2
, we show the existence of a finitely generated regular extension of
k
k
over which
G
G
admits outer automorphisms. Using this result and a construction of groups of type
A
\mathsf {A}
from groups of type
D
\mathsf {D}
, we construct new examples of groups of type
2
A
n
^2\mathsf {A}_n
with
n
≡
3
mod
4
n\equiv 3\bmod 4
and the first examples of type
2
A
n
^2\mathsf {A}_n
with
n
≡
1
mod
4
n\equiv 1\bmod 4
(
n
≥
5
)
(n\geq 5)
that are not
R
R
-trivial, hence not rational (nor stably rational).