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*][1], Helmer Aslaksen, Eng-Chye Tan, Chen-bo Zhu Pacific J. Math. 168(2): 207-215 (1995).

However, you can see this with one copy of $\mathrm{SO}(2,\mathbb{F})$ already, 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 (the group is abelian).  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*][2], Sean Lawton & Adam S. Sikora 
Algebras and Representation Theory volume 20, pages 1133–1141(2017).


  [1]: https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-168/issue-2/Invariant-theory-of-special-orthogonal-groups/pjm/1102620558.full
  [2]: https://arxiv.org/pdf/1604.02164.pdf