  In this paper, we prove four-moment theorems for multidimensional free
Poisson limits on free Wigner chaos or the free Poisson algebra. We prove that,
under mild technical conditions, a bi-indexed sequence of free stochastic
integrals in free Wigner algebra or free Poisson algebra converges to a free
sequence of free Poisson random variables if and only if the moments with order
not greater than four of the sequence converge to the corresponding moments of
the limit sequence of random variables. Similar four-moment theorems hold when
the limit sequence is not free, but has a multidimensional free Poisson
distribution with parameters $\lambda>0$ and $\alpha=\{\alpha_i: 0\ne
\alpha_i\in \mathbb{R}, i=1, 2, \cdots\}$.
