The Jarque-Bera test is based of the convergence, if the vector is i.i.d. normal random variables, of
$$\texttt{skewness}(x) \rightarrow \mathcal{N}(0, 6) ; \texttt{kurtosis}(x) \rightarrow \mathcal{N}(3, 24)$$
and moreover, both are asymptotically independent. Then, we have the statistic
$$J = \frac{n}{6}\left(S^2 + \frac{(K-3)^2}{4}\right) \rightarrow \chi^{2}(2)$$