In this wonderful lecture note on Clifford Algebra and Spin(N) Representations,  Somehow I find some inconsistency with his Tables of Euclidean and Minkowski fermions, compared with the Table of Wikipedia. (see the bottom ) Wikipedia page shows that for Euclidean fermions, even 6d (6,0) and 7d (7,0) have Majorana fermions. But his Table says NO Majorana in both 6d (6,0) and 7d (7,0).  Wikipedia page shows that for Minkowski fermions, even 8d (7,1) and 1d (0,1) or 9d (8,1)  have Majorana fermions. But his Table says NO Majorana in both 8k and 8k+1 d.  Could any expert clarify why there is a discrepancy above? In summary, in Wikipedia, out of mod 8 dimensions, Wikipedia shows that 5 out of 8 cases have Majorana fermions, in both Euclidean and Minkowski signatures.  only said that 3 out of 8 cases have Majorana fermions, in both Euclidean and Minkowski signatures. Here I marked the red colors for the inconsistency data of  compared with Wikipedia Table. See Wikipedia page :
Almost all standard sources on the existence of Majorana spinors (e.g. Appendix B.1 to Polchinski's "String Theory", Vol. 2) do so in a way I consider inherently ugly:   A priori, we are dealing with an irreducible complex representation $(V,\rho)$ of the Clifford algebra of signature $(p,q)$, i.e. generalized Dirac spinors. That Majorana spinors exist means abstractly that there is a  on $V$, i.e. a conjugate-linear map $\phi : V\to V$ with $\phi^2 = \mathrm{id}_V$ that commutes at least with the $\mathfrak{so}(p,q)$ action.  Every single source I can find for Majorana spinors uses operations like the transpose, complex conjugation and Hermitian adjoint on the $\Gamma$-matrices to obtain matrices acting on the same space. This is abstractly wrong, the transpose acts on the dual, the complex conjugation on the conjugate, and the Hermitian adjoint needs an inner product we have no reason for choosing. Of course, since $V$ is finite-dimensional, one can pick a basis and define the uncanonical isomorphisms to its dual and its conjugate, but I find this inelegant, particularly since the standard derivations require us to make a particular such choice with respect to the signs the $\Gamma$-matrices have under e.g. ${}^\dagger$. Finally, the Majorana spinors are usually defined by some equation involving an unnatural and arbitrary-looking product of $\Gamma$-matrices, which varies from source to source according to different sign conventions and sign choices made in the course of the derivation.  It's inelegant because the rest of the theory of spinors can be developed without making such uncanonical choices. Both the uniqueness of the dimension of the irreducible Dirac representations (there are two of them in odd dimensions) and the existence of the Weyl spinors in even dimensions can be derived purely from the abstract properties of the Clifford algebra, no choices made, no transpose, adjoint or conjugate occuring. The (admittedly slightly subjective question) is: Is there a way to show in which dimensions Majorana spinors exist that neither requires an uncanonical choice of basis nor arbitrary choices of signs?  Some partial results:   In even dimensions, the Dirac representation is necessarily self-conjugate since it is the only irreducible representation of the Clifford algebra, so all that is left to show is that a conjugate-linear $\mathfrak{so}$-equivariant map on it squares to $\mathrm{id}_V$ and not to $-\mathrm{id}_V$. However, I can't seem to exhibit any particular equivariant map on it that one could simply check for its square. In odd dimensions, one first needs to figure out whether the two inequivalent Dirac representations are conjugate to each other or self-conjugate.     As further motivation that a clear proof using only canonical properties of the Clifford algebra itself is required, consider the confusing and contradictory claims in the literature:   Polchinski, "String Theory", Vol. 2, p.434: $\mathrm{SO}(d-1,1)$ has Majoranas for $d = 0,1,2,3,4 \mod 8$, corresponding to a special case of $p-q = d-1-1 = 6,7,0,1,2 \mod 8$. Fecko, "Differential Geometry and Lie Groups for Physicists", pp. 651: $\mathrm{Cliff}(p,q)$ has Majoranas for $p-q = 0,2\mod 8$.  This clearly conflicts with Polchinski's claims e.g. for $d=3$. Figueroa-O'Farrill, , pp. 18: We have Majoranas for $p-q = 0,6,7\mod 8$ and "symplectic Majoranas" for $p-q = 2,3,4\mod 8$.   Note that these results conflict simply by the number of possible $p-q$ regardless of whether I've correctly taken care of the differing conventions of whether $p$ or $q$ denotes timelike dimensions.
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