Suppose we have two (multivariate) random variables $X_1\in \mathbb{R}^{X_1}$, $X_2\in \mathbb{R}^{X_2}$ that are uncorrelated (zero covariance, or you may think that are spanning two orthogonal spaces in $\mathcal{L}^2$). Suppose the existence up to the fourth moment (for the formula see Mardia 1970) of such variables.
Can we infer any implications/restrictions starting from the null covariance to any of such higher moments of the individual variables (or linear combination/product of them)?
[If it helps, please consider the univariate case.]
In general you cannot say anything on the moments of individual random vectors $X_1$ and $X_2$.
The general property that holds for uncorrelated random variables is that you can easily compute the variance of the linear combination of these random vectors, e.g. for the univariate case you have $$ \mathbb{V}ar(\alpha X_1 + \beta X_2) = \alpha^2 \mathbb{V}ar(X_1) + \beta^2 \mathbb{V}ar(X_2). $$
For the higher moments its more complicated and nothing can be claimed in the general case.
There are some special cases where you can have nice properties, i.e. when $X_1, X_2 \sim \mathcal{N}(\mu, \Sigma)$, then the uncorrelation implies also independence of $X_1$ and $X_2$, if $\Sigma \approx \sigma^2 I_p$.