Constructing groups of Type E7 with certain Tits Index

Question

In a new survey on $E_8$, namely

Skip Garibaldi - E8 the most exceptional group

, the author gives an example (Example 8.4., page 15) on how to construct a group of type E8 with a prescribed Tits-Index. This construction was actually invented by Tits and bears his name..

From the inclusion of $F_4 \subset E_6 \subset E_8$, after applying $H^1(k,-)$, a isotropic group of type $E_8$ with semi simple anisotropic kernel of type $E_6$ arises from an Albert Algebra $A$, which is a $F_4$ torsor (meaning an element of $H^1(k,F_4)$). (Note that this example serves to explain some properties of the Rost invariant of this $E_8$, but we want to focus on the construction itself.)

We will remember this for later and call it #Above

In the Paper

J. Tits - Stronger inner anisotropic forms of simple algebraic groups

,on the last page the author gives an example of a group of type $E_7$ having semi simple anisotropic kernel $D_5 \times A_1$, or in Tits's notation is of index $E^{48}_{7,1}$, meaning the vertex $6$ is circled.

He writes:

"..[27, Proposition 5] shows that groups with this index are classified by anisotropic quadratic forms $q$ in $10$ variables, whose invariant $c(q)$ is a division quaternion algebra."

1.Does he mean classified as constructed like in #Above ?

If he means like in #Above:

1.1.Is a group with that index constructed by choosing such a $q$, looking at $SO(q)$, which is of type $D_5$ and adjoint, noticing that $D_5\subset D_6 \subset E_7$ and applying $H^1(k,-)$?

What is confusing to me is that the anisotropic kernel of the resulting $E_7$ is not just $D_5$ but $D_5 \times A_1$. But on the other hand $D_5$ can never occur as anisotropic kernel, so it might all fit together.

2.What is known about the quadratic forms $q$?

I ponder they have splitting pattern $(1,2,2)$, after analyzing the more isotropic Tits-Indexes and taking into account Vishik's classification of splitting patterns.

Answer 1

This might shed some light on relationship between anisotropic quadratic forms in 10 variables and the desired forms of $E_7$, though it uses results more recent than Tits, and doesn't quite answer your questions as stated. Bruce Allison worked out the following results in his paper "Structurable division algebras and relative rank one simple Lie algebras" in Lie Algebras and Related Topics: Proceedings of a Summer Seminar Held June 26-July 6, 1984. More polished and easier to find is his later paper on "Tensor products of composition algebras, Albert Forms and Some Exceptional Simple Lie Algebras" in Transactions of the American Mathematical Society, Vol. 306, No. 2 (Apr., 1988)

Assume the base field $k$ does not have characteristic $2$ nor $3$. Take a division quaternion algebra $B$ and a division octonion algebra $C$ over $k$, for which $A = B \otimes C$ is a division algebra too. Equivalently, $B$ and $C$ contain no common quadratic extension of $k$. This tensor product $A = B \otimes C$ is a 32-dimensional algebra with involution -- a special sort of algebra that Allison calls a structurable algebra. Its skew elements form a 10-dimensional subspace $S$, and (with the division algebra condition), the norm form on $S$ is anisotropic. This is the $q$, and $q$ determines $A$ up to isomorphism (as a $k$-algebra with involution) -- see Theorem 5.4 of Allison's later paper.

By construction, we can describe the $q$ in more detail. The norm forms on $B$ and $C$ are Pfister forms, say $<1,b_1> \otimes <1,b_2>$ and $<1,c_1> \otimes <1,c_2> \otimes <1,c_3>$, respectively. The form $q$ is the orthogonal sum of the skew subspace of $B$ and (negative) the skew subspace of $C$. So we have $$q = (b_1, b_2, b_1 b_2, -c_1, -c_2, -c_3, -c_1 c_2, -c_2 c_3, - c_3 c_1, -c_1 c_2 c_3).$$ It may be easy to describe the possible $q$, using the classification of quaternion and octonion algebras.

Allison uses the structurable algebra $A$ to construct a 5-graded Lie algebra $K_{-2} \oplus K_{-1} \oplus K_0 \oplus K_1 \oplus K_2$ of the desired type. $K_{\pm 1}$ are 32-dimensional, identified with $A$. $K_{\pm 2}$ are 10-dimensional, identified with $S$. Lie brackets on $K_{\pm 1}$ arise from the structurable algebra via $[a_1, a_2] = a_1 \bar a_2 - a_2 \bar a_1$. $K_0$ is harder to describe.

Allison proves that all Lie algebras of type $E_{7,1}^{48}$ arise from this construction, and by the end of his latter paper (Transactions 1988) he constructs examples, e.g., over a field $R((T_1, T_2))$ of Laurent series.