  We study the image of the $\ell$-adic Galois representations associated to
the four vector valued Siegel modular forms appearing in the work of Chenevier
and Lannes. These representations are symplectic of dimension $4$. Following a
method of Dieulefait, we determine the primes $\ell$ for which these
representations are absolutely irreducible. In addition, we show that their
image is "full" for all primes $\ell$ such that the associated residual
representation is absolutely irreducible, except in two cases.
