Let $K$ be a $k$-algebra, consider $o(n,K)$ the set of orthogonal matrices of size $n^2$ over $K$ ( i.e $M M^T = I $). Let´s do the case $n=2$. Let's consider the ring $$ A = k\left[ {x,y,z,w} \right]/\left( {x^2 + y^2 - 1,xz + yw,z^2 + w^2 - 1} \right). $$
Where the system is from the equation $M M^T = I$ with $M$ a matrix with entries $x$, $y$, $z$, $w$ , and then equaling to zero. I have to prove that there exist a natural bijection between:
$o(2,K)$ and $\operatorname{Hom}_k( A , K )$ where $\operatorname{Hom}_k( A , K )$ denotes the homomorphism of $k$-algebras $f\colon A \to K$ i.e homomorphism of rings, that fix $k$. Only guessing I think that the map it´s that it take a matrix, and send to the homomorphism, that send each coordinate $ T_i $ of $A$, to one of the coordinates of the orthogonal matrix.
But I can't prove neither injectivity nor surjectivity.
EDITED: Thanks!