  We classify the holomorphic Borcherds products of singular weight for all
simple lattices of signature $(2,n)$ with $n \geq 3$. In addition to the
automorphic products of singular weight for the simple lattices of square free
level found by Dittmann, Hagemeier and the second author, we obtain several
automorphic products of singular weight $1/2$ for simple lattices of signature
$(2,3)$. We interpret them as Siegel modular forms of genus $2$ and explicitly
describe them in terms of the ten even theta constants. In order to rule out
further holomorphic Borcherds products of singular weight, we derive estimates
for the Fourier coefficients of vector valued Eisenstein series, which are of
independent interest.
