On the $\mathbb{A}^{1}$-Euler characteristic of the variety of maximal tori

$\mathbb{A}^{1}$-Euler characteristic is an invariant from the motivic homotopy theory which associates to a smooth algebraic variety over a field $k$ a non-degenerate symmetric bilinear form over $k$ (more precisely, an element of the Grothendieck–Witt ring of symmetric bilinear forms over $k$). This invariant generalizes the topological Euler characteristic in the sense that over the field of complex numbers it recovers the topological Euler characteristic of the manifold of complex points. In the talk I will recall the construction of this invariant and give an overview of some computations. Then I will discuss what is known about the $\mathbb{A}^{1}$-Euler characteristic of the variety of maximal tori in a reductive group and present the corresponding generalized splitting principle.