Let $E/F$ be a quadratic extension of number fields and $V,W$ are hermitian and skew hermitian vector space over $E$ whose dimension is $n,m$ respectively.

Let $\pi$ be a irreducible tempered cuspidal representation of $U(V)(F)$ and $\theta(\pi)$ is the theta lift to $U(W)(F)$ with some fixed choice of character $\chi:A_E^{\times}/E^{\times} \to S^1$ whose restriction to $A_F^{\times}/F^{\times}$ corresponds to the quadratic character of Galois group $Gal(E/F)$ by class field theory.

Suppose that $\theta(\pi)$ is non-vanishing. Then can we say the theta lift $\theta(\theta(\pi))$ to $U(V)$ also non-vanishing?

If is does, then what is $\theta(\theta(\pi))$? I strongly guess it should be $\pi$. Is it right?

Thanks.