Soit $P=a+b+c+d$, et $S=a-b-c+d$. \[ Q=a+b-c-d, \quad R=a-b+c-d \] Balculer \[ T=P Q\left(P^{2}+Q^{2}\right)-R S\left(R^{2}+S^{2}\right) \]

Step 1 ：Let's denote $P=a+b+c+d$, $Q=a+b-c-d$, $R=a-b+c-d$, and $S=a-b-c+d$.

Step 2 ：We are asked to calculate $T=P Q\left(P^{2}+Q^{2}\right)-R S\left(R^{2}+S^{2}\right)$.

Step 3 ：Substitute the expressions for $P$, $Q$, $R$, and $S$ into the equation for $T$ and simplify.

Step 4 ：After simplifying, we find that $T = 16a^3b + 16ab^3 - 16c^3d - 16cd^3$.

Step 5 ：So, the final expression for $T$ is \(\boxed{16a^3b + 16ab^3 - 16c^3d - 16cd^3}\).