Let
G
(
m
)
G(m)
denote
S
U
(
m
)
{\rm {SU}}(m)
or
S
p
(
m
)
{\rm {Sp}}(m)
. It is shown that when
m
≥
5
G
(
m
)
m \geq 5\,G(m)
cannot act smoothly on
W
n
,
2
W_{n,2}
, the complex Stiefel manifold of orthonormal
2
2
-frames in
C
n
\mathbf C^n
, for
n
n
odd, with connected principal isotropy type equal to the class of maximal tori in
G
(
m
)
G(m)
. This demonstrates an important difference between
W
n
,
2
W_{n,2}
,
n
n
odd, and
S
2
n
−
3
×
S
2
n
−
1
S^{2n-3}\times S^{2n-1}
in the behavior of differentiable transformation groups. Exactly the same holds for
S
O
(
m
)
{\rm {SO}}(m)
or Spin
(
m
)
(m)
if it is further assumed that a maximal
2
2
-torus of
S
O
(
m
)
{\rm {SO}}(m)
has fixed points.
2
^{2}