  We provide a much shorter but even more powerful proof of an algebraic
identity, which can be used to establish the direct and the converse inequality
under Type IV superorthogonality. As an application, we obtain the optimal
order of the formal constant in the direct inequality. This order turns out to
be also sharp for Type III superorthogonality. When $p\geq 2$, our result
recovers the optimal order of the constant in the Burkholder-Gundy inequality
in martingale theory, and also recovers the currently best order of the
constant in the reverse Littlewood-Paley inequality. We also introduce variants
of Type IV superorthogonality, under which we prove the converse inequality.
