In the paper "Motivic construction of cohomological invariants", the author displays a list of known norm varieties for several $n,p$ on page $11$. For $p=2, n=5$ it says that a norm variety is given by the variety of singular trace zero lines in a reduced Albert algebra with cohomological invariant $f_5$, which is a twist of $F_4/P_4$. We call this thing $X$.

Question: Where do I find a reference?

To break this down a bit.

The Chow-Ring and anisotropic motivic decomposition of $F_4/P_4$ is well known. It consists of eight Tate-twisted copies of the same $(3,3)$ generalized Rost-motive.

$f_5$ gives a $2$ Rost-motive (of dimension $15$) and thus a binary summand.

So the interesting thing for me to realize is that twisting gives varieties with totally different motivic summands.

On the other hand there is obviously another norm variety for the symbol of $f_5$, namely a minimal $5$-Pfister-neighbour. So this gives us another example of non isomorphic varieties with isomorphic motives (i assume the decomposition of $X$ is the same like a norm quadric) or at least some isomorphic summands.



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