For some Lie groups you will need additional invariants. For example, for the even orthogonals you will need the Pfaffian in addition to the Trace.
See for example:
Invariant theory of special orthogonal groups, Helmer Aslaksen, Eng-Chye Tan, Chen-bo Zhu Pacific J. Math. 168(2): 207-215 (1995).
However, you can already see this with one copy of $\mathfrak{so}(2,\mathbb{F})$ and $\mathrm{SO}(2,\mathbb{F})$ acting by conjugation. For in that case the Lie algebra is $\left\{\left(\begin{array}{cc}0&a\\-a&0\end{array}\right)|\ a\in \mathbb{F}\right\}$ and the conjugation action is trivial. In this case the trace is identically 0 and the Pfaffian is the only useful invariant.
In case you are interested, the Lie group version of this question was addressed by myself and Sikora here:
Varieties of Characters, Sean Lawton & Adam S. Sikora Algebras and Representation Theory volume 20, pages 1133–1141(2017).