Let
X
X
be a complete smooth variety defined over a number field
K
K
and let
i
i
be an integer. The absolute Galois group
G
a
l
K
\mathrm {Gal}_K
of
K
K
acts on the
i
i
th étale cohomology group
H
e
´
t
i
(
X
K
¯
,
Q
ℓ
)
H^i_{\mathrm {\acute {e}t}}(X_{\bar K},\mathbb {Q}_\ell )
for all primes
ℓ
\ell
, producing a system of
ℓ
\ell
-adic representations
{
Φ
ℓ
}
ℓ
\{\Phi _\ell \}_\ell
. The conjectures of Grothendieck, Tate, and Mumford-Tate predict that the identity component of the algebraic monodromy group of
Φ
ℓ
\Phi _\ell
admits a reductive
Q
\mathbb {Q}
-form that is independent of
ℓ
\ell
if
X
X
is projective. Denote by
Γ
ℓ
\Gamma _\ell
and
G
ℓ
\mathbf {G}_\ell
respectively the monodromy group and the algebraic monodromy group of
Φ
ℓ
s
s
\Phi _\ell ^{\mathrm {ss}}
, the semisimplification of
Φ
ℓ
\Phi _\ell
. Assuming that
G
ℓ
0
\mathbf {G}_{\ell _0}
satisfies some group theoretic conditions for some prime
ℓ
0
\ell _0
, we construct a connected quasi-split
Q
\mathbb {Q}
-reductive group
G
Q
\mathbf {G}_{\mathbb {Q}}
which is a common
Q
\mathbb {Q}
-form of
G
ℓ
∘
\mathbf {G}_\ell ^\circ
for all sufficiently large
ℓ
\ell
. Let
G
Q
s
c
\mathbf {G}_{\mathbb {Q}}^{\mathrm {sc}}
be the universal cover of the derived group of
G
Q
\mathbf {G}_{\mathbb {Q}}
. As an application, we prove that the monodromy group
Γ
ℓ
\Gamma _\ell
is big in the sense that
Γ
ℓ
s
c
≅
G
Q
s
c
(
Z
ℓ
)
\Gamma _\ell ^{\mathrm {sc}}\cong \mathbf {G}_{\mathbb {Q}}^{\mathrm {sc}}(\mathbb {Z}_\ell )
for all sufficiently large
ℓ
\ell
.