Around the Grothendieck–Serre conjecture on principal $G$-bundles

The following result due to M. Ojanguren illustrates a well-known conjecture due to J.-P. Serre and A. Grothendieck.

Theorem. Let $k$ be a field and $X$ be a $k$-smooth irreducible affine variety over $k$ (the characteristic of $k$ is not $2$). Let $E$ and $F$ be two quadratic spaces over the regular function ring $k[X]$. If $E$ and $F$ are isomorphic over the fraction field $k(X)$, then they are isomorphic locally for the Zariski topology.
The conjecture asserts, particularly, that a similar statement holds for principal $G$-bundles, where $G$ is any reductive algebraic group over the field $k$.
Many examples illustrating the conjecture will be presented and a “final” result will be discussed.