Two questions on the Grothendieck ring of varieties 1) In the definition of the Grothendieck ring of varieties over a field $k$, which definition of the various notions of "variety" is chosen? Finite type and separated, or maybe more?
2) If $\mathbb{L}$ is the class of the affine line in the Grothendieck ring, does the class of $\mathrm{GL}_n$ equal $(\mathbb{L}^n - 1) \cdot \dotsc \cdot (\mathbb{L}^n - \mathbb{L}^{n-1})$? Somehow this should be true, but for this I would need a "fibration relation", which does not seem to follow from the scissor relation.

From what I've read, it seems like separated and finite type are fairly standard. Poonen, in his paper "The Grothendieck Ring of Varieties is not a Domain" adds geometrically reduced- I haven't seen this other places, but I haven't looked too hard.
For the second, there does exist a fibration relation- seeing it is difficult without going to more complicated categories with isomorphic Grothendieck rings. Consider the inclusion $\mathrm{Var}_k\to\mathrm{Space}_k$- this gives rise to a homomorphism of grothendieck rings, which turns out to be an isomorphism. The following argument then works in the category of algebraic spaces over $k$, and the relation in fact holds in the Grothendieck ring of varieties:
Consider a fibration $E\to B$ with fiber $F$. In the category of algebraic spaces, there exists an open set $U\subset B$ such that $E_U\to U$ is isomorphic to $U\times F\to U$. Let  $V= B\setminus U$, a closed subset of $B$. So we have $[E]=[F][U]+[E_V]$, and by noetherian induction, we may say that $[E_V]=[F][V]$, and so $[E]=[F][B]$. Now you can apply the argument you seem to have in mind in (2).
