Consider Grothendieck's ring of varieties, $K\_0(\mathcal{Var}\_k)$ over a field $k$, and the
Lefschetz element $\mathbb{L}=[\mathbb{A}^1\_k]$.
Consider also the subring $\mathcal{L}\_k\subset K\_0(\mathcal{Var}\_k)$ consisting of all classes which are polynomials in $\mathbb{L}$, which for convenience I'll call the "Lefschetz ring".


An algebraic group $G$ over $k$ is called *special* (as defined by Serre and classified by Grothendieck) if any principal $G$-bundle over a $k$-variety is locally trivial in the Zariski topology.


It is known that $GL\_n(k)$ is in $\mathcal{L}\_k$, for example: $$[GL\_2(k)]=(\mathbb{L}^2-1)(\mathbb{L}^2-\mathbb{L}).$$ Also, $\mathbb{G}\_a\cong \mathbb{A}^1\_k$ and $\mathbb{G}\_m\cong k^\times$ are clearly in $\mathcal{L}\_k$. Since the examples above are all special groups, (and not having found references with computation of the classes in $K\_0(\mathcal{Var}\_k)$ of orthogonal groups (which are not special, for example)), I wonder if all special groups have their classes in the Lefschetz ring. 


More generally, the question is: is there any known relation between special groups and groups whose classes are in $\mathcal{L}\_k$ (if necessary, one could assume $k=\mathbb C$)?



Serre's special groups and polynomials in Lefschetz element
Seems that class of any split reductive algebraic group, special or not, lies in this ring. Use $[G]=[G/B][B]$ and Bruhat decomposition for $G/B$ (while $B$ is glued from ${\mathbb G}\_a$'s and ${\mathbb G}\_m$'s).


