  We give an effective procedure to explicitly find the decomposition of a
polarized abelian variety into its simple factors if a period matrix is known.
Since finding this datum is not easy, we also provide two methods to compute
the period matrix for a polarized abelian variety, depending on the given
geometric information about it. These results work particularly well in
combination with our previous work on abelian varieties with group actions,
since they allow us to fully decompose such varieties by successively
decomposing their factor subvarieties, even when these no longer have a group
action. We highlight that we do not require to determine the full endomorphism
algebra of any of the (sub)varieties involved. We illustrate the power of our
algorithms with two byproducts: we find a completely decomposable Jacobian
variety of dimension 101, filling this Ekedahl-Serre gap, and we describe a new
completely decomposable Jacobian variety of CM type of dimension 11.
