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This chapter summarizes the different cases about Moufang quadrangles of type E⁶, E₇ and E₈. The first case is that the building at infinity of the Bruhat-Tits building Ξ‎ is an unramified quadrangle; the second, a semi-ramified quadrangle; and the third, a ramified quadrangle. The chapter considers a theorem that takes into account two root group sequences, both of which are either indifferent or the various dimensions, types, etc., are as indicated in exactly one of twenty-three cases. It also presents a number of propositions relating to a quaternion division algebra and a quadratic space of type Eℓ for ℓ = 6, 7 or 8. Finally, it emphasizes the fact that the quadrangles of type F₄ could have been overlooked in the classification of Moufang polygons.

This chapter deals with the case that the building at infinity of the Bruhat-Tits building Ξ‎ is a Moufang quadrangle of type E⁶, E₇, and E₈. It begins with a hypothesis that takes into account a quadratic space of type Eℓ for ℓ = 6, 7 or 8, K which is complete with respect to a discrete valuation, the two residues of Ξ‎, and the two root group sequences of a Moufang polygon. It then considers the case that Ξ‎ is an unramified quadrangle if the proposition δ‎_Ψ‎ = 2 holds. It also explains two other propositions: Ξ‎ is a semi-ramified quadrangle if δ‎_Λ‎ = 1 and δ‎_Ψ‎ = 2 holds, and a ramified quadrangle if δ‎_Λ‎ = δ‎_Ψ‎ = 1 holds.

This chapter deals with the residues of a Bruhat-Tits building whose building at infinity is an exceptional quadrangle. It begins with the remark that if Λ‎ is an arbitrary quadratic space of type Eℓ for ℓ = 6, 7 or 8 or of typeF₄ over a field K that is complete with respect to a discrete valuation, and if in the F4-case the subfield F is closed with respect to this valuation and if Δ‎ is the corresponding Moufang quadrangle of type Eℓ or F₄, then there always exists a unique affine building Ξ‎ such that Δ‎ is the building at infinity of Ξ‎ with respect to its complete system of apartments. The chapter also considers the standard embedding of the apartment A in the Euclidean plane which takes the intersection of A and R to the set of eight triangles containing the origin. Finally, it describes a Moufang polygon with two root group sequences.

This chapter proves various results about Moufang quadrangles. It first considers the notions of a proper involutory set, a proper indifferent set, and a proper anisotropic pseudo-quadratic space. It then shows that the root group sequence Ω‎ is isomorphic to a root group sequence of exactly one of six types relating to some proper involutory set, some non-trivial anisotropic quadratic space, some proper indifferent set, some proper anisotropic pseudo-quadratic space, and some quadratic space. It also describes the degree of a finite purely inseparable field extension as a power of the characteristic, an isomorphism from a root group sequence of Δ‎ to the Moufang quadrangle, and abelian and non-abelian groups.

This chapter presents various results about quadratic forms of type F₄. The Moufang quadrangles of type F₄ were discovered in the course of carrying out the classification of Moufang polygons and gave rise to the notion of a quadratic form of type F₄. The chapter begins with the notation stating that a quadratic space Λ‎ = (K, L, q) is of type F₄ if char(K) = 2, q is anisotropic and: for some separable quadratic extension E/K with norm N; for some subfield F of K containing K² viewed as a vector space over K with respect to the scalar multiplication (t, s) ↦ t²s for all (t, s) ∈ K x F; and for some α‎ ∈ F* and some β‎ ∈ K*. The chapter also considers a number of propositions regarding quadratic spaces and discrete valuations.

This chapter deals with the case that the building at infinity Λ‎ of the Bruhat-Tits building Ξ‎ is a Moufang quadrangle of type F₄. It begins with the hypothesis stating that Λ‎ = (K, L, q) is a quadratic space of type F₄, K is complete with respect to a discrete valuation ν‎ and F is closed with respect to ν‎, Λ‎ is the Moufang quadrangle corresponding to a root group sequence, and R₀ and R₁ as the two residues of Ξ‎. The chapter also considers the theorem supposing that Λ‎ is of type F₄ and that R₀ and R₁ are not both indifferent, and claims that both cases really occur. Finally, it presents the proposition that R₀ and R₁ are both indifferent if and only if q is totally wild.

This chapter summarizes the results about the residues of Bruhat-Tits buildings other than those associated with the exceptional Moufang quadrangles examined in previous chapters. It first considers cases, for which it assumes that Λ‎ is complete with respect to a discrete valuation in an appropriate sense. It then presents the Coxeter diagram of Ξ‎ and the vertex set S of such diagram, along with the J-residue of the building Ξ‎, which is called a gem if J is the complement in the S of a special vertex. The chapter also discusses the structure of the gems of Ξ‎ as well as cases in which the pseudo-quadratic space is defined to be ramified or unramified.