This is a question about Kallenberg's textbook, Foundations of Modern Probability, theorem 11.25. Let $I_n$ denote the Weiner-Ito stochastic integral on $L^2([0,1]^n,\lambda^n)$, with $\eta$ an isonormal Gaussian process. Therefore, we have that $I_n(\otimes_{i \leq n} h_i) = \prod_{i \leq n} \eta(h_i)$, for $h_i$ orthogonal elements of separable Hilbert space, $H = L^2([0,1],\lambda)$.
Now, the theorem is showing (1): $$I_n(\bigotimes_{j \leq m}e_j^{\otimes n_j})=\prod_{j \leq m}p_{n_j}(\eta e_j)$$ where $(e_j)$ are distinct orthonormal elements of $H$ with "multiplicities" $n_j$ sum to $n$, and $p_n$ is the $n^{th}$ Hermite polynomial (if you are referring to Wikipedia, it is the probabilist's Hermite polynomial), defined as an orthonormal basis wrt. the standard normal distribution on $\mathbb{R}$.
Question:
My question is about a remark, stating this is equivalent to showing (2):
$$I_n(\bigotimes_{j \leq m}h_j^{\otimes n_j})=\prod_{j \leq m} I_{n_j}h_j^{\otimes n_j}$$
for orthonormal $h_1...h_k \in H$, where $I_n(h^{\otimes n}) = ||h||p_n(\eta \hat{h})$, where $\hat{h} = h/||h||$,
I am unable to see why these are equivalent. In particular, knowing (1), how can we derive (2)? What I've tried is consider the simplest case, so assuming (1), let's consider $h^{\otimes n} = \sum c_i \bigotimes_{j}e_{i,j}^{n_{i,j}}$ (i.e., representation in basis). Then, I derived that $I_n(h^{\otimes n}) = \sum_i c_i I_n(\bigotimes_{j}e_{i,j}^{n_{i,j}}) =_{1} \prod_j p_{n_{i,j}}(\eta e_{i,j})$, and I'm not sure at all how to go from here to showing the claim.