Invariants of symmetric matrices

$\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\eS}{\mathscr{S}}$ $\DeclareMathOperator{SO}{SO}$  $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\eQ}{\mathscr{Q}}$ $\DeclareMathOperator{\tr}{tr}$

Fix an integer $m>2$, and denote by $\eS_m$  the  space of real, symmetric   $m\times m$ matrices. The group  $\SO(m)$ of  orthogonal transformations  of $\bR^m$ acts on $\eS_m$ by conjugation. Fix a unit vector  $\eta\in\bR^m$. We obtain a subgroup   $\SO(m-1)$ of $\SO(m)$ consisting of transformations that fix the vector $\eta$.   Denote by $\eQ_m$ the space  of   $\SO(m-1)$-invariant  homogeneous quadratic polynomials  on $\eS_m$.   Then  $\eQ_m$ is spanned by the quadratic polynomials

$$   A\mapsto \tr A^2,\;\; A\mapsto (\tr A)^2, \;\;A\mapsto |A\eta|^2, $$

$$ A\mapsto (A\eta,\eta)^2,\;\; A\mapsto (\tr A)(A\eta,\eta), $$

where   $(-,-)$ denotes the canonical inner product on $\bR^m$.

There is   a simple proof  of this fact due to Robert Bryant.