Theorem. The actual compact \(\operatorname {Spin}(8)\) has outer Lie-automorphism group \(S_3\) [fcap-003C]

✍️sourceAGENTDRAFTED

Let \(Q_8^+\) be the positive quadratic form on \(\mathbb R^8\), and let \[G=\operatorname {Spin}(Q_8^+)\longrightarrow SO(8)\] be the actual vector-generated, reverse-norm-one carrier and its vector action \(\pi \). The concrete Lie-group and covering results of Proposition [fcap-0037] and Proposition [fcap-003A] make \(G\) compact, connected, and simply connected, with \(\pi \) its two-sheeted universal cover. The even-word sign-change map of Proposition [fcap-0039] identifies this positive-square carrier with the compact negative-square convention used in the classical source, compatibly with the vector action.

Let \(\nu _+\) and \(\nu _-\) be the two real eight-dimensional half-spin representations. Their complexifications have highest weights \(\omega _4\) and \(\omega _3\), so the comparison of Proposition [fcap-003B] and the exterior construction of Theorem [fcap-002Y] identify them, up to chosen intertwiners, with the two concrete complex half-spin modules. Average a positive definite inner product over compact \(G\). In an orthonormal basis either representation gives \[\nu :G\longrightarrow O(8).\] Connectedness puts its image in \(SO(8)\). Since \(G\) is simply connected, the universal cover \(\pi \) has a unique based lift \[\Theta :G\longrightarrow G,\qquad \pi \circ \Theta =\nu .\] The two maps \(\Theta (xy)\) and \(\Theta (x)\Theta (y)\) are based lifts on the connected space \(G\times G\), so uniqueness makes \(\Theta \) a group homomorphism.

Its derivative is nonzero because the half-spin representation has nonzero weights. The kernel of the derivative is an ideal of the simple Lie algebra \(\mathfrak {so}_8\), hence it is zero. Equal dimensions make \(\Theta \) a local Lie-group isomorphism. Its image is open, and compactness makes it closed; connectedness of \(G\) makes the image all of \(G\). Thus \(\Theta \) is a covering of a simply connected target by a connected source and is a Lie-group automorphism. Let Lie-group automorphisms act on representation classes by inverse pullback, \[\varphi \mathbin {\cdot }[M]=[M\circ \varphi ^{-1}].\] With this left-action convention, \(\Theta ^{-1}\) sends the vector class to the chosen half-spin class because \(\pi \circ \Theta =\nu \).

Every Lie-group automorphism permutes the three classes. After inner conjugation it preserves a chosen maximal torus and positive Weyl chamber; its differential therefore induces a based \(D_4\) root-datum automorphism. The diagram calculation of Proposition [fcap-003B] shows that it permutes precisely the outer-node weights \(\omega _1,\omega _3,\omega _4\). Inner automorphisms act trivially on isomorphism classes, so inverse pullback defines a homomorphism \[\operatorname {Out}_{\mathrm {Lie}}(G)\longrightarrow S_3.\] There is a second kind of actual Lie-group automorphism. Conjugation by any unit vector in the Clifford algebra preserves even vector products and reverse norm and has conjugation by its inverse as inverse. On the vector module it gives orthogonal conjugation of \(\pi \); on the odd Clifford action it interchanges \(S^+\) and \(S^-\). Its permutation fixes the vector class and swaps the two half-spin classes. Together with the lifted half-spin automorphism, these permutations generate \(S_3\).

This permutation action has inner kernel. Suppose an automorphism \(\varphi \) fixes all three representation classes. Equivalence of \(\pi \circ \varphi \) with \(\pi \) gives an intertwiner; after polar normalization it is a matrix \(A\in O(8)\). If \(\det A=1\), lift \(A\) to \(g\in G\). After composing \(\varphi \) with the inverse inner automorphism \(\operatorname {Int}(g)\), its projection agrees with \(\pi \). The pointwise discrepancy from the identity lies continuously in \(\ker \pi =\{1,-1\}\), hence is identically \(1\) on connected \(G\). The remaining automorphism is the identity. If \(\det A=-1\), first compose with unit-vector conjugation. The result is inner by the preceding case, but unit-vector conjugation swaps the two half-spin classes, contradicting the hypothesis. Therefore \[\operatorname {Out}_{\mathrm {Lie}}(G)\simeq S_3.\] A suitable composition of the two constructed outer classes maps to a three-cycle and therefore has order three in the outer group. This does not show that the chosen half-spin lift \(\Theta \) has order three, and no representative has been normalized here to cube to the identity. No splitting \(S_3\to \operatorname {Aut}_{\mathrm {Lie}}(G)\) is asserted.

[adams1996exceptional, Chapter 5, Theorem 5.6, pp. 34--35] proves the compact group theorem and the permutation of its three eight-dimensional representations. [fulton2004representation, Section 20.3, pp. 312--314] independently describe the corresponding complex vector/half-spin triality and root diagram. The carrier translation, universal-cover lift, surjectivity, and kernel argument above place that comparison on the actual FCAP Spin group.

The pinned declaration TauCeti.D4Tripled.trialityAutomorphism has literal cube one on a tripled toral-closure carrier. Its source file does not identify that carrier with the actual group \(G\). Accordingly it is not used to claim cube one for \(\Theta \) or as a formal badge for this theorem.