  In our previous work, he second and the third named authors constructed the
Ikeda type lift for the exceptional group $E_{7,3}$ from an elliptic modular
cusp form. In this paper, we prove an explicit formula for the period or the
Petersson norm of the Ikeda type lift in terms of the product of the special
values of the symmetric square $L$-function of the elliptic modular form.
  There are similar works done by the first author with his collaborator, but
new technical inputs are required and developed to overcome some difficulties
coming from the hugeness of $E_{7,3}$.
