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The semisimple groups over a local field have been classified by Tits, cf. [1] "Classification of algebraic semisimple groups" in Boulder and [2] "Reductive groups over local fields" in Corvallis. In both references, Tits gives the construction of the semisimple groups corresponding to each index, except for certain groups of exceptional type. I am mostly interested in the nonquasi-split groups of exceptional type, that is, using the names in [2, §4] and the indices from [1, Table II]:

  • $^3\mathrm{E}_6$ which has index $^1\mathrm{E}_{6,2}^{16}$, and

  • $^2\mathrm{E}_7$ which has index $\mathrm{E}_{7,4}^9$

Tits states that these are constructed by means of a central division algebra $D$ of degree $3$ and $2$ respectively, but I cannot find a good reference. Does anyone know where to look for?

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    $\begingroup$ I suspect that Tits does not give constructions of $^3\mathrm{E}_6$ and $^2\mathrm{E}_7$ because there is no nice constructions! $\endgroup$
    – Mikhail Borovoi
    Commented Jun 20, 2019 at 23:56
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    $\begingroup$ Let $G_0$ be the corresponding simply connected split group, and $G_0^{\rm ad}$ be the adjoint group. Let $\eta(D)\in H^2(k,\mu_d)$ denote the class of $D$ (where $\mu_d=Z(G_0)$ and $d$ is 3 or 2, respectively). Then one should somehow use a construction of Martin Kneser (see also the book by Platonov and Rapinchuk) in order to lift $\eta$ to a cohomology class $\xi\in H^1(k,G_0^{\rm ad})$. Here one should use the assumption that $k$ is a local field. $\endgroup$
    – Mikhail Borovoi
    Commented Jun 21, 2019 at 0:09
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    $\begingroup$ Let $c\in Z^1(k,G_0^{\rm ad})$ be a representative of $\xi$. Then it is not hard to twist $G_0$ using $c$, and you obtain the desired group $_c G_0$ of type $^3\mathrm{E}_6$ or $^2\mathrm{E}_7$, respectively, and its faithful $k$-irreducible representation $V$ over $k$ with centralizer $D$ in ${\rm End}_k(V)$. $\endgroup$
    – Mikhail Borovoi
    Commented Jun 21, 2019 at 0:18

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For $E_7$ there is at least a nice construction at the level of Lie algebra. It is so called Tits (or Tits-Freudenthal) construction that takes a Jordan algebra $J$ and a quaternion algebra $Q$ and produces a Lie algebra of type $E_7$ by some explicit formulas. In your situation you specialize $J$ to the split Albert algebra. The exact reference is

J. Tits, Algebres alternatives, algebres de Jordan et algebres de Lie exceptionnelles. I. Construction, Nederl. Akad. Wetensch. Proc. Ser. A 69 = Indag. Math. 28 (1966), 223–237.

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