We prove that the Kalton centralizer on
L
p
[
0
,
1
]
L_p[0,1]
, for
0
>
p
>
∞
0>p>\infty
, is not strictly singular: in all cases there is a Hilbert subspace on which it is trivial. Moreover, for
0
>
p
>
2
0>p>2
there are copies of
ℓ
q
\ell _q
, with
p
>
q
>
2
p>q>2
, on which it becomes trivial. This is in contrast to the situation for
ℓ
p
\ell _p
spaces, in which the Kalton-Peck centralizer is strictly singular.