$\newcommand{\eS}{\mathscr{S}}$ $\DeclareMathOperator{\SO}{SO}$ $\newcommand{\eP}{\mathscr{P}}$ $\newcommand{\bR}{\mathbb{R}}$ $\DeclareMathOperator{\tr}{tr}$ Let $m>1$ be an integer and denote by $\eS_m$ the vector space of symmetric $m\times m$ real matrices. The group $\SO(m)$ acts by conjugation on $\eS_m$. The space of $\SO(m)$-invariant quadratic polynomials on $\eS_m$ is spanned by the two polynomials

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

Let $\SO(m-1)$ be the subgroup of $\SO(m)$ consisting of orthogonal transformations of $\bR^m$ that fix a unit vector $\eta$.

What is the space of $\SO(m-1)$-invariant on $\eS_m$. More precisely, can one explicitly write a basis of this space?

Clearly $\tr A^2$ and $(\tr A)^2$ are such polynomials, and so are

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

where $(-,-)$ denotes the natural inner product on $\bR^m$. Are there any more $\SO(m-1)$ invariant quadratic polynomials? (I am inclined to believe that the above is the complete list.) Update After Robert Bryant's answer I lost my initial inclination.

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# Invariants of symmetric matrices

$\newcommand{\eS}{\mathscr{S}}$ $\DeclareMathOperator{\SO}{SO}$ $\newcommand{\eP}{\mathscr{P}}$ $\newcommand{\bR}{\mathbb{R}}$ $\DeclareMathOperator{\tr}{tr}$ Let $m>1$ be an integer and denote by $\eS_m$ the vector space of symmetric $m\times m$ real matrices. The group $\SO(m)$ acts by conjugation on $\eS_m$. The space of $\SO(m)$-invariant quadratic polynomials on $\eS_m$ is spanned by the two polynomials

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

Let $\SO(m-1)$ be the subgroup of $\SO(m)$ consisting of orthogonal transformations of $\bR^m$ that fix a unit vector $\eta$.

What is the space of $\SO(m-1)$-invariant on $\eS_m$. More precisely, can one explicitly write a basis of this space?

Clearly $\tr A^2$ and $(\tr A)^2$ are such polynomials, and so are

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

where $(-,-)$ denotes the natural inner product on $\bR^m$. Are there any more $\SO(m-1)$ invariant polynomials? (I am inclined to believe that the above is the complete list.)