You can multiply with a diagonal matrix that is congruent to the identity and get the determinant equal to one.

Please correct the subscripts in the second equation.

What is $\mathbb{R}^+$? Is $f(x)=0$ allowed?

doi.org/10.1016/0021-8693(92)90189-S

If a reductive group is split over $\mathbb{F}_p$ and acts as such on a module, then a root subgroup fixes a weight vector if and only if the $\mathbb{F}_p$ points of the root subgroup fix the vector.

Compare mathoverflow.net/questions/350590/….

It is definitely the case that Freudenthal invented suspension and named it Einhängung. He referred to it as the `big fish' that he had caught. When asked what mathematical result he was most proud of, this was his answer.

@Mohan You really should explain what you use about the ring. For instance, it would not work when $A$ is a polynomial ring in three variables over the reals.

Not getting through the preliminary selection was humbling. Handling disappointments is crucial in research.

One may use Nagao's Theorem. Especially when the field has only two elements, as the easy way is then blocked.

Related to mathoverflow.net/questions/176393/git-over-integers/…

The hypothesis that $G_{\mathbb{C}}$ is reductive looks too weak. One would prefer to have $G$ reductive over $S$ in the sense of SGA3. That is, one wants $G$ to be smooth over $S$ with geometric fibers that are connected reductive.