Central Elements of the Universal Enveloping Algebra and Functions of Matrix Elements

We construct central elements in $U(\mathfrak{o}_N)$ and $U(\mathfrak{g}_2)$ by using the first main theorem of invariant theory. We also construct new functions of matrix elements; these functions naturally arise when describing the center of $U(\mathfrak{g}_2)$, just as Pfaffians arise when describing the center of $U(\mathfrak{o}_N)$.