A group
G
G
is representable in a Banach space
X
X
if
G
G
is isomorphic to the group of isometries on
X
X
in some equivalent norm. We prove that a countable group
G
G
is representable in a separable real Banach space
X
X
in several general cases, including when
G
≃
{
−
1
,
1
}
×
H
G \simeq \{-1,1\} \times H
,
H
H
finite and
dim
X
≥
|
H
|
\dim X \geq |H|
, or when
G
G
contains a normal subgroup with two elements and
X
X
is of the form
c
0
(
Y
)
c_0(Y)
or
ℓ
p
(
Y
)
\ell _p(Y)
,
1
≤
p
>
+
∞
1 \leq p >+\infty
. This is a consequence of a result inspired by methods of S. Bellenot (1986) and stating that under rather general conditions on a separable real Banach space
X
X
and a countable bounded group
G
G
of isomorphisms on
X
X
containing
−
I
d
-Id
, there exists an equivalent norm on
X
X
for which
G
G
is equal to the group of isometries on
X
X
.
We also extend methods of K. Jarosz (1988) to prove that any complex Banach space of dimension at least
2
2
may be renormed with an equivalent complex norm to admit only trivial real isometries, and that any complexification of a Banach space may be renormed with an equivalent complex norm to admit only trivial and conjugation real isometries. It follows that every real Banach space of dimension at least
4
4
and with a complex structure may be renormed to admit exactly two complex structures up to isometry, and that every real Cartesian square may be renormed to admit a unique complex structure up to isometry.