Working on a recent question led me to the following invariant-computation problem : let
$$ A=\bigg\lbrace P \in {\mathbb Q}[X_1,X_2,X_3,X_4] \ \bigg| \\ \quad\ P(X_1X_3+X_2X_4+X_1X_4,\ X_2X_3,\ X_1X_3+X_2X_4+X_2X_3,\ X_1X_4)=\\ =P(X_1,X_2,X_3,X_4) \bigg\rbrace $$
Then $A$ is a sub algebra of ${\mathbb Q}[X_1,X_2,X_3,X_4]$. Is it trivial ? Does anyone know how to find a finite generating set for it ?