Denote $C$ as complex number and $R=C[a,b,c]$ polynomial ring with 3 variables. Denote $ax^2+2bxy+cy^2$ corresponding quadratic form associated to $a,b,c$. Let $O(2)$ acting on $(x,y)$ to induce action on $a,b,c$ via quadratic form $ax^2+2bxy+cy^2$ where identify $(x,y)$ as coordinate of $R^2$ plane. Denote $S$ as $O(2)$ invariant part of $R$. Write $b^2=E+ac, T=a+c$.
If $f(a,c)\in R$ is $O(2)$ invariant, then there is a rotation by $\pi/4$ s.t. $(a,c)\to (1/2(a+c),1/2(a+c))=(1/2 T,1/2 T)$.(Since there is no dependence on $b$, that part of transformation is not seen.) Hence $f(a,c)$ is a polynomial purely in $T$.
For any $f(a,b,c)\in R$, it is clear that $b$ must be quadratic via $diag\{1,-1\}$ action. $f(a,b,c)=g(a,b^2,c)=g(a,E+ac,c)=\sum_i E^i f_i(a,c)$ with $f_i\in R$. Furthermore, one notes that $O(2)$ transformation is homogeneous. Hence it suffices to assume $f$ is homogeneous of degree $n$. Now consider $f_0(a,c)$ as I am expecting this is invariant under $O(2)$ action. $f$ is $O(2)$ invariant and $E$ is $O(2)$ invariant. Since $O(2)$ preserve degree of polynomial, it must be the case $f_0(a,c)$ is invariant as every element of transformed $f_0(a,c)$ is homogeneous degree $n$ whereas other $f_{i>1}(a,c)$ are homogeneous of degree less than $n$. Hence $f_0$ is a polynomial in $T$. Therefore I see $f(a,b,c)-f_0(a,c)=E\times h(a,b,c)$ with $h$ lower degree. Hence $f$ is an element of $C[T,E]$ which shows $C[a,b,c]^{O(2)}=C[T,E]$.
$\textbf{Q1:}$ Is above idea correct? Since my idea is very ad hoc, is there a way to systematize it?
$\textbf{Q2:}$ This is only mentioned in Mukai's Intro to invariants and moduli pg 3's proof of Prop 1.1. The book simply says $C[a,b,c]$'s $O(2)$ invariant is generated by $E,T$. (It seems that book is implying this is trivial or obvious.) How so or am I missing some important ingredient?