In Exercises 17.17 and 17.18 of Jech's Set Theory book, he shows that if the language $L_\kappa$ satisfies the Weak Compactness Theorem, then $\kappa$ is a weakly inaccessible cardinal. Also, in Theorem 17.13 of the same book, it is shown that if $\kappa$ is strongly inaccessible, then $L_\kappa$ satisfies the Weak Compactness Theorem if and only if $\kappa$ is weakly compact.
This seems to suggest that it is consistent that there is a cardinal $\kappa$ for which $L_\kappa$ satisfies the Weak Compactness Theorem but such that $\kappa$ is only weakly inaccessible and not strongly inaccessible.
Can anyone provide a reference for such a consistency result?