Here is an extract of the doctoral thesis of C. Lewis under the supervision of D. Joyce (<https://people.maths.ox.ac.uk/joyce/theses/LewisDPhil.pdf>, 1998):




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**2.6 Spin Bundles and the Dirac Operator**


To consider spin bundles over a $Spin(7)$ manifold $M$, it is usually best to first consider Clifford algebras.


Let $V$ be a finite dimensional vector space with an inner product defined
upon it. Let $e\_1$, $e\_2$, ... , $e\_n$ be an orthonormal basis for $V$.


Then the Clifford algebra, $C\_n$, of $V$ is defined to be the algebra generated by the elements $e\_1$, $e\_2$, ... , $e\_n$ subject to the relations


$$e\_i^2 =-1$$,
$$e\_ie\_j + e\_je\_i = 0\text{ for }i \neq j$$


Considered as a vector space $C\_n$ is of dimension $2n$, spanned by elements of the form


$$e\_1^{\delta\_1}e\_2^{\delta\_2}\cdots e\_n^{\delta\_n}$$


where $\delta\_i = 0$ or 1.


Now consider the case $n = 8$. In this case it can be shown that


$$C\_8 = \mathbb{R} (16),$$


the algebra of $16\times 16$ matrices with values in $\mathbb{R}$. [Sal, p.171]


*Look also [here](https://mathoverflow.net/questions/194279/explicit-isomorphism-between-cl8-and-mathbbr16)*


Thus we may consider $\mathbb{R}$ (16) as a $C\_8$ module.


We may define the group $Spin(8)$ as the subset of $C\_8$ consisting of all even products $x\_1x\_2\cdots x\_{2r-1}x\_{2r}$ of elements of $V$, with each $\|x\_i\| = 1$. (Similarly we might have defined $Spin(7)$ as the subset of $C\_7$ consisting of all even products $x\_1x\_2\cdots x\_{2r-1}x\_{2r}$ of elements of $\mathbb{R}^8$, with each $\|x\_i\| = 1$.)


Now let us consider the element $v = e\_1e\_2\cdots e\_8$ of $C\_8$. Then $v$ is a involution of $C\_8$, and commutes with every element of $Spin(8)$, and hence $\mathbb{R}^{16}$ splits as a $Spin(8)$ module into the eigenspaces of $v$.


Thus $\mathbb{R}^{16}=\Delta\_+\oplus\Delta\_-$, where $\Delta\_+$ is the $+1$ eigenspace of $v$, and $\Delta\_-$ is the $-1$ eigenspace of $v$. We call them the positive and negative spin representations of $Spin(8)$.


Now suppose that $M$ is a $Spin(7)$ manifold. Then we have that $M$ is a spin manifold i.e. there exists a spin structure of $M$, a principal $Spin(8)$ bundle $\tilde{E}$ covering the $SO(8)$ bundle of frames for the tangent bundle.


Now since we have a principal $Spin(8)$ bundle, and the two $Spin(8)$ modules
(namely $\Delta\_+$ and $\Delta\_-$), we may form two vector bundles associated to the principal spin bundles by means of the two spin representations.


We call these bundles $S\_+$ and $S\_-$, the positive and negative spinor bundles, and their sections are known as positive and negative spinors. It is perhaps worth noting at this point that the group $Spin(7)$ is the subgroup of $SO(8)$ preserving a spinor, and hence the manifold $M$ will possess a constant spinor. Thus we will have isomorphisms $S\_+ \equiv\Lambda^0\otimes \Lambda^2\_7$ and $S\_-\equiv \Lambda^1$.




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The question is: **Why is it true that $S\_+ \equiv\Lambda^0\otimes \Lambda^2\_7$ and $S\_-\equiv \Lambda^1$?**


Any suggestion is welcome.



Isomorphisms of Positive and Negative Spinor Bundles
You're really asking an algebra question about how the various representations of $\mathrm{Spin}(8)$ interact. There are lots of places where you can read about this, but [here](http://www.math.duke.edu/~bryant/Spinors.pdf "here") is a set of notes that I wrote, *Remarks on Spinors in Low Dimensions*, that explains this and that some people have found useful.


Once you have read the first few pages, you'll see why there is a canonical 'nondegenerate' bilinear pairing $S\_-\otimes S\_+ \longrightarrow T$, where $T$ is the tangent bundle of $M$. Then, once you have a unit-size positive spinor $s$, i.e., a section of $S\_+$ of unit size, then $S\_+$ splits as the line bundle $\mathbb{R}s$ and its orthogonal $V$, which is a $7$-plane bundle. The map above restricted to $S\_-\simeq S\_-\otimes \mathbb{R}s\longrightarrow T$ then becomes an isomorphism, so $S\_-$ is isomorphic to the tangent bundle $T$. Then, considering the map $S\_-\otimes V\to T$ and the fact that all three of these bundles are irreducible under $\mathrm{Spin}(7)$, i.e., the stabilizer of $s$ (which also is proved in the above reference), you'll get, by duality, an embedding of $V$ into $T\otimes S\_- = T\otimes T$, and, again using the above notes, you'll see that this goes into $\Lambda^2(T)\subset T\otimes T$ and must be perpendicular to the $\mathrm{Spin}(7)$-irreducible subbundle ${\frak{spin}}(7) = \Lambda^2\_{21}\subset \Lambda^2(T)$. By definition $\Lambda^2\_7$ is the orthogonal complement to $\Lambda^2\_{21} $ in $ \Lambda^2(T)$.


