  The paper investigates the locus of non-simple principally polarised abelian
$g$-folds. We show that the irreducible components of this locus are
$\Is^g_{D}$, defined as the locus of principally polarised $g$-folds having an
abelian subvariety with induced polarisation of type $D=(d_1,\ldots,d_k)$,
where $k\leq\frac{g}{2}$. The main theorem produces Humbert-like equations for
irreducible components of $\Is^g_{D}$ for any $g$ and $D$. Moreover, there are
theorems which characterise the Jacobians of curves that are \'etale double
covers or double covers branched in two or four points.
