Theta blocks related to root systems

Gritsenko, Skoruppa and Zagier associated to a root system $R$ a theta block ${\vartheta }_R$, which is a Jacobi form of lattice index. We classify the theta blocks ${\vartheta }_R$ of $q$-order $1$ and show that their Gritsenko lift is a strongly-reflective Borcherds product of singular weight, which is related to Conway’s group ${\mathrm{Co}}_0$. As a corollary we obtain a proof of the theta block conjecture by Gritsenko, Poor and Yuen for the pure theta blocks obtained as specializations of the functions ${\vartheta }_R$.