The definition of pure spinor that I've found in the mathematical literature (refs 2 and 3) allows vectors to have complex components. But in the physics literature, I've seen the same idea applied to Clifford algebras over the real numbers instead. Ref 4 is an example. The following definition covers both cases. (This definition seems to be consistent with the definition in chapter 3 of ref 1, but I'm not a mathematician, so please consult the references to double-check what I've written here.)

Let $\gamma_1,\gamma_2,...,\gamma_N$ be a set of generators for the Clifford algebra Cliff($p,q$), satisfying $$ \newcommand{\bfx}{\mathbf{x}} \gamma_j \gamma_k + \gamma_k \gamma_j=2g_{jk} $$ as usual, where $g_{jk}$ is a flat metric with signature $(p,q)$. A linear combination $\bfx = \sum_k x_k\gamma_k$ with real coefficients $x_k$ will be called a vector. Or, we can allow the coefficients to be complex, in which case the signature $(p,q)$ doesn't matter except for the total number of dimensions $N\equiv p+q$. In either case, a subspace $S$ such that $\bfx^2=0$ for all $\bfx\in S$ will be called a totally isotropic subspace, and it will be called a maximal totally isotropic subspace if it has the largest possible number of dimensions among all such subspaces. A spinor $\psi$ will be called pure if $\bfx \psi=0$ for all $\bfx\in S$ for some maximal totally isotropic subspace $S$. This assumes that spinors — not necessarily pure — are already defined in the standard way, as in the first paragraph above.

Now the question can be reframed like this: Are all spinors pure? Or, for what values of $N$ (in the complex case) and what signatures $(p,q)$ (in the real case) are all spinors pure?

  I don't know the complete answer, but here are a couple of excerpts indicating that spinors are not always pure:

• According to ref 4, regarding the case of Lorentzian signature: "In 3,4,6 and 10 dimensions, a general spinor in the smallest irreducible spinor representation of the Lorentz group is automatically pure." The fact that normed division algebras exist only in dimensions 1, 2, 4, and 8 seems to be related to this. The relationship is reviewed in ref 5 and the appendix of ref 6.

• According to section 109 in ref 2: "It is obvious that any spinor can be taken, in an infinity of ways, as a sum of pure spinors..."

• From page 109 in ref 1: "In general not all spinors will be pure; whereas we can always choose a basis of pure spinors, linear combinations of pure spinors will not in general be pure."

• From page 113 in ref 1: "Eight dimensions are interesting as the lowest number of dimensions in which not all semi-spinors are pure." (I think what they call semi-spinors is what physicists call chiral or Weyl spinors, which matches the definition above in the even-dimensional case.)

Compared to the first excerpt, the last three excerpts assume either a different signature or a different field (complex instead of real). That's why the conclusions differ.

Pure spinors: definition

Let $g_{jk}$ be a flat metric with signature $(p,q)$. The definition of pure spinor that I've found in the mathematical literature (refs 2 and 3) either allows vectors to have complex components, which includes the case illustrated in the question, or focuses on signatures for which $|p-q|\leq 1$. But in the physics literature, I've seen the same idea applied to Clifford algebras over the real numbers with Lorentzian signature instead. Ref 4 is an example. The following definition covers both cases. This definition seems to be consistent with the definition in chapter 3 of ref 1, but I'm not a mathematician, so please consult the references to double-check what I've written here.

Let $\gamma_1,\gamma_2,...,\gamma_N$ be a set of generators for the Clifford algebra Cliff($p,q$), satisfying $$ \newcommand{\bfx}{\mathbf{x}} \gamma_j \gamma_k + \gamma_k \gamma_j=2g_{jk} $$ as usual. A linear combination $\bfx = \sum_k x_k\gamma_k$ with real coefficients $x_k$ will be called a vector. Or, we can allow the coefficients to be complex, in which case the signature $(p,q)$ doesn't matter except for the total number of dimensions $N\equiv p+q$. In either case, a subspace $S$ such that $\bfx^2=0$ for all $\bfx\in S$ will be called a totally isotropic subspace, and it will be called a maximal totally isotropic subspace if it has the largest possible number of dimensions among all such subspaces. A spinor $\psi$ will be called pure if $\bfx \psi=0$ for all $\bfx\in S$ for some maximal totally isotropic subspace $S$. This assumes that spinors — not necessarily pure — are already defined in the standard way, as in the first paragraph above.

When are all spinors pure?

Now the question can be reframed like this: For what dimensions (in the complex case) and what signatures (in the real case) are all spinors pure? I don't know the complete answer, but here are a couple of excerpts regarding special cases:

• According to ref 4, regarding the special case of Lorentzian signature: "In 3,4,6 and 10 dimensions, a general spinor in the smallest irreducible spinor representation of the Lorentz group is automatically pure." The fact that normed division algebras exist only in dimensions 1, 2, 4, and 8 seems to be related to this. The relationship is reviewed in ref 5 and the appendix of ref 6.

Regarding the cases typically considered in the mathematical literature (complex coefficients, or real coefficients with $|p-q|\leq 1$):

• According to section 109 in ref 2: "It is obvious that any spinor can be taken, in an infinity of ways, as a sum of pure spinors..."

• From page 109 in ref 1: "In general not all spinors will be pure; whereas we can always choose a basis of pure spinors, linear combinations of pure spinors will not in general be pure."

• From page 113 in ref 1: "Eight dimensions are interesting as the lowest number of dimensions in which not all semi-spinors are pure." (What they call semi-spinors is what physicists would call chiral or Weyl spinors in the even-dimensional case.)

Altogether, there are dimensions/signatures where all spinors are pure (like the case shown in the question), and dimensions/signatures where not all spinors are pure.

Why are pure spinors interesting?

I haven't studied this subject enough to know all of the reasons, but one reason is that their association with maximal totally isotropic subspaces provides pure spinors an appealing geometric flavor. This seems to be useful for proving some properties of general spinors, at least in the cases of complex coefficients or $|p-q|\leq 1$ that are usually studied in the mathematical literature.

With Lorentzian signature, pure spinors play an important role in the study of supersymmetry, as illustrated in refs 4,5,6.

By the way, ref 4 defines a pure spinor to be one that satisfies the Cartan-Penrose equation $$ \sum_{j,k}g^{jk}(\overline\psi\gamma_j\psi)\gamma_k\psi=0. $$ This looks different than the definition I showed above, but we can see the relationship between the definitions by writing $\bfx\equiv \sum_{j,k}g^{jk}(\overline\psi\gamma_j\psi)\gamma_k$, so that the preceding equation is $\bfx\psi=0$. Multiply this on the left by $\bfx$ so see that it is consistent with $\bfx^2=0$, and note that a maximal totally isotropic subspace is one-dimensional when the signature is Lorentzian (that is, when $p$ or $q$ equals $1$).

Other uses of the name "pure spinor"

• According to ref 4: "In 3,4,6 and 10 dimensions, a general spinor in the smallest irreducible spinor representation of the Lorentz group is automatically pure." The fact that normed division algebras exist only in dimensions 1, 2, 4, and 8 seems to be related to this. The relationship is reviewed in ref 5 and the appendix of ref 6.

• According to section 109 in ref 2: "It is obvious that any spinor can be taken, in an infinity of ways, as a sum of pure spinors..."

• From page 113 in ref 1: "Eight dimensions are interesting as the lowest number of dimensions in which not all semi-spinors are pure." (I think what they call semi-spinors is what physicists call chiral or Weyl spinors, which matches the definition above in the even-dimensional case.)

• According to ref 4, regarding the case of Lorentzian signature: "In 3,4,6 and 10 dimensions, a general spinor in the smallest irreducible spinor representation of the Lorentz group is automatically pure." The fact that normed division algebras exist only in dimensions 1, 2, 4, and 8 seems to be related to this. The relationship is reviewed in ref 5 and the appendix of ref 6.

• According to section 109 in ref 2: "It is obvious that any spinor can be taken, in an infinity of ways, as a sum of pure spinors..."

• From page 109 in ref 1: "In general not all spinors will be pure; whereas we can always choose a basis of pure spinors, linear combinations of pure spinors will not in general be pure."

• From page 113 in ref 1: "Eight dimensions are interesting as the lowest number of dimensions in which not all semi-spinors are pure." (I think what they call semi-spinors is what physicists call chiral or Weyl spinors, which matches the definition above in the even-dimensional case.)

Compared to the first excerpt, the last three excerpts assume either a different signature or a different field (complex instead of real). That's why the conclusions differ.