Consider the action of $G = {\rm SO}(2)$ by conjugation on real symmetric matrices, that is: $$g\cdot A = gAg^{-1} = gAg^{t},$$ for $A$ a symmetric real matrix $2 \times 2$. The author says that I can see the orbit space $$\frac{{\rm Sym}(2,\mathbb{R})}{{\rm SO}(2)} = \frac{\mathbb{R}^{2}}{S_{2}} =\frac{\mathbb{R}}{S_{2}}\times\mathbb{R},$$ where $S_{2}$ denotes the symmetric group of 2 letters, and the last splitting is valid since $S_{2}$ acts trivially on the line $\mathbb{R}(1,1)$. Here, the first parcel of the splitting is the subspace of diagonal matrices with trace equals $0$. My question is, why is the first parcel necessarily the subspace of traceless matrices?
Thanks for every tip!