Independence between products of Gaussian random variables

Question

Let $$X \sim \mathcal{N}(0, 1), \quad Y\sim \mathcal{N}(0, 1), \quad Z \sim \mathcal{N}(0, 1).$$

Let $A_1 = XY$ and $A_2 = XZ.$

I can show that $A_1$ and $A_2$ are uncorrelated. Also, as per this question I know their PDF. Are $A_1$ and $A_2$ independent?

Answer 1

I think the answer is no. Indeed $$E[A_1^2A_2^2] = E[X^4]E[Y^2]E[Z^2] = 3$$ however $$E[A_1^2]=E[A_2^2] = 1$$