Triality with even and odd spinors The spin group with mixed metric signs ($p$ for positive and $q$ for negative) is usually defined as
$$
Spin(p,q) = Pin(p,q) \cap C\ell^{+}_{p,q}\,,
$$
and in the special case of $Spin(8,0)$ one has the phenomenon of Triality. Defined this way, one can only use objects of the Clifford algebra with an even number of elements.
My questions is, why is this always defined with even elements? And can one obtain the Triality automorphism with both even and odd elements of the Clifford algebra?
Thanks!
 A: The key is the Cartan-Dieudonne theorem. It says every linear isometry of $\mathbb{R}^n$ is expressible as a product of $\le 2n$ hyperplane reflections. (That number of reflections is even or odd depending on if the isometry is orientation-preserving or reversing, respectively.) Indeed, every hyperplane is the orthogonal complement of some vector, and the composition of the reflections across $u^\perp$ and $v^\perp$ will act trivially on $\mathrm{span}\{u,v\}^\perp$ and as a rotation by $2\theta$ in the 2D plane $\mathrm{span}\{u,v\}$, where $\theta$ is the angle between $u$ and $v$. The (real version of the) spectral theorem implies every orthogonal transformation in some basis is block-diagonal with $2\times2$ rotation matrix blocks (and possibly a $\pm1$ in the corner in odd dimension); I leave it as an exercise to see how this is related.
Given a real inner product space $V$, the Clifford algebra $C\ell(V)$ is the quotient of the tensor algebra $TV$ by the relations $v^2=-\|v\|^2$, or equivalently $uv+vu=-2\langle u,v\rangle$ (after polarization). If we consider the standard coordinate vector space $V=\mathbb{R}^n$ with standard basis $\{e_i\}$, then $C\ell(n)$ is the free unital associative $\mathbb{R}$-algebra generated by $n$ anticommuting square roots of $-1$, namely $\{e_i\}$, generalizing the quaternions $C\ell(2)=\mathbb{H}$. Of course, quaternions give 3D rotations (and 4D, incidentally), so there seems to be a mismatch in these numbers!
Quaternions model 3D rotations with conjugation, but in the spirit of spinors being the "square root of geometry" (Atiyah), we could instead view conjugation as giving reflections.
The idea is that $V\subset C\ell(V)$ is a subspace which is closed under conjugation by its own elements - in fact, $-uvu^{-1}$ is the reflection of $v$ across $u^\perp$ (which you can verify by decomposing $v$ into parallel and perpendicular components wrt $u$), and so conjugating by an even number of (wlog) unit vectors yields rotations, by Cartan-Dieudonne. Thus, we define $\mathrm{Spin}(V)$ to be generated by products of evenly-many unit vectors in $C\ell(V)$, which is contained in the even subalgebra $C\ell(V)_0$.
All spin groups $\mathrm{Spin}(n)$ ($n>1$) have an outer "duality" automorphism coming from conjugating by a single unit vector. In $\mathrm{SO}(n)$, that corresponds to conjugating by reflections (which are not inner automorphisms of $\mathrm{SO}(n)$). But in dimension $8$, we have $\mathrm{Out}\,\mathrm{Spin}(8)=S_3$, so the duality automorphism corresponds to a $2$-cycle, but there is a nontrivial $3$-cycle automorphism! (The other $3$-cycle would be its inverse, and the other two $2$-cycles would be products of the duality automorphism with the two $3$-cycles.) Such permutations only describe automorphisms of $\mathrm{Spin}(8)$ up to inner automorphism, of course.
Actually constructing a triality typically involves octonions. I find it easier to see what's explicitly happening at the lie algebra level. Let $\mathbb{O}$ be the octonions, an 8D real algebra, with $\mathrm{Im}\mathbb{O}$ the 7D subspace of purely imaginary octonions. For any pure imaginary octonion $\mathbf{u}$, define the multiplication maps $L_u(x)=ux$ and $R_u(x)=xu$. The automorphisms of $\mathbb{O}$ as an algebra form a $14$-dimensional compact Lie group called $G_2$. We can check that $\mathfrak{so}(8)=L_{\mathrm{Im}\,\mathbb{O}}\oplus R_{\mathrm{Im}\,\mathbb{O}}\oplus\mathfrak{g}_2$. Any permutation in $S_3$ can be turned into an automorphism of $\mathfrak{so}(8)$ by fixing $\mathfrak{g}_2$ and permuting $\{L_u,R_u,-L_u-R_u\}$ accordingly (in the same way for each pure imaginary octonion $u$). Indeed, with the Killing form on $\mathfrak{so}(n)$, we can find that $\{L_u,R_u,-L_u-R_u\}$ are at $120^\circ$ angles from each other for every $u$, and $\{L_u,R_u\}$ are orthogonal to $\{L_v,R_v\}$ whenever $u$ and $v$ are orthogonal octonions.
There is also another way to create triality by realizing $\mathrm{Spin}(8)$ as a subgroup of $\mathrm{SO}(8)^3$ acting on isotopies, or equivalently trilinear forms (also called trialities). Maybe I'll add in details of this later, or link it if I've already written it somewhere, but it's probably available in something written by John Baez or Robert Bryant.
