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\begin{document}

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\section*{SO(8)}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{SubgroupLattice}{Subgroup lattice}\dotfill \pageref*{SubgroupLattice} \linebreak
\noindent\hyperlink{homotopy_groups}{Homotopy groups}\dotfill \pageref*{homotopy_groups} \linebreak
\noindent\hyperlink{cohomology_of_classifying_spaces}{Cohomology of classifying spaces}\dotfill \pageref*{cohomology_of_classifying_spaces} \linebreak
\noindent\hyperlink{structure_and_exceptional_geometry}{$G$-Structure and exceptional geometry}\dotfill \pageref*{structure_and_exceptional_geometry} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{subgroup_lattice_2}{Subgroup lattice}\dotfill \pageref*{subgroup_lattice_2} \linebreak
\noindent\hyperlink{cohomology}{Cohomology}\dotfill \pageref*{cohomology} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Among all [[special orthogonal groups]] $SO(n)$, the case of $SO(8)$ is special, since in the [[ADE classification]] of [[simple Lie groups]] it corresponds to [[D4]], which makes its [[representation theory]] enjoy \emph{[[triality]]}.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{SubgroupLattice}{}\subsubsection*{{Subgroup lattice}}\label{SubgroupLattice}

\begin{prop}
\label{Spin7SubgroupsInSpin8}\hypertarget{Spin7SubgroupsInSpin8}{}
\textbf{([[Spin(7)]]-[[subgroups]] in [[Spin(8)]])}

There are precisely 3 [[conjugacy classes of subgroups|conjugacy classes]] of [[Spin(7)]]-[[subgroups]] inside [[Spin(8)]], and the [[triality]] group $Out(Spin(8))$ [[action|acts]] [[transitive action|transitively]] on these three classes.

$\backslash$begin\{center\} $\backslash$begin\{xymatrix\} $\backslash$mathrm\{Spin\}(7) $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}dr{\tt \symbol{94}}\{$\backslash$iota'\} $\backslash$ \& $\backslash$mathrm\{Spin\}(8) \& $\backslash$mathrm\{Spin\}(7) $\backslash$ar@\{\_\{(\}-{\tt \symbol{62}}\}l{\tt \symbol{94}}-\{$\backslash$iota\} $\backslash$ $\backslash$mathrm\{Spin\}(7) $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}ur\emph{-\{ $\backslash$iota'` \} $\backslash$end\{xymatrix\} $\backslash$end\{center\}}

\end{prop}
(\hyperlink{Varadarajan01}{Varadarajan 01, Theorem 5 on p. 6}, see also \hyperlink{Kollross02}{Kollross 02, Prop. 3.3 (1)})

\begin{prop}
\label{G2IsIntersectionOfSpin7SubgroupsInSpin8}\hypertarget{G2IsIntersectionOfSpin7SubgroupsInSpin8}{}
\textbf{([[G2]] is [[intersection]] of [[Spin(7)]]-[[subgroups]] of [[Spin(8)]])}

The [[intersection]] inside [[Spin(8)]] of any two [[Spin(7)]]-subgroups from distinct [[conjugacy classes of subgroups]] (according to Prop. \ref{Spin7SubgroupsInSpin8}) is the [[exceptional Lie group]] [[G2]], hence we have [[pullback squares]] of the form

$\backslash$begin\{center\} $\backslash$begin\{xymatrix\} G\_2 $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}r $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}d \& $\backslash$mathrm\{Spin\}(7) $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}d{\tt \symbol{94}}-\{$\backslash$iota{\tt \symbol{94}}$\backslash$prime\} $\backslash$ $\backslash$mathrm\{Spin\}(7) $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}r\emph{-\{$\backslash$iota\} \& $\backslash$mathrm\{Spin\}(8) $\backslash$end\{xymatrix\} $\backslash$end\{center\}}

\end{prop}
(\hyperlink{Varadarajan01}{Varadarajan 01, Theorem 5 on p. 13})

\begin{prop}
\label{}\hypertarget{}{}
We have the following [[commuting diagram]] of [[subgroup]] inclusions, where each [[square]] exhibits a [[pullback]]/[[fiber product]], hence an [[intersection]] of subgroups:

$\backslash$begin\{center\} $\backslash$begin\{xymatrix\} $\backslash$mathrm\{SU\}(2) $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}r $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}d $\backslash$ar@\{\}dr|-\{ $\backslash$mbox\{ $\backslash$tiny (pb) \} \} \& $\backslash$mathrm\{SU\}(3) $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}r $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}d $\backslash$ar@\{\}dr|-\{ $\backslash$mbox\{ $\backslash$tiny (pb) \} \} \& $\backslash$mathrm\{G\}\_2 $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}r $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}d $\backslash$ar@\{\}dr|-\{ $\backslash$mbox\{ $\backslash$tiny (pb) \} \} \& $\backslash$mathrm\{Spin\}(7) $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}d{\tt \symbol{94}}\{B $\backslash$iota'\} $\backslash$ $\backslash$mathrm\{Spin\}(5) $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}r \& $\backslash$mathrm\{Spin\}(6) $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}r \& $\backslash$mathrm\{Spin\}(7) $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}r\emph{-\{$\backslash$iota\} \& $\backslash$mathrm\{Spin\}(8) $\backslash$end\{xymatrix\} $\backslash$end\{center\}}

Here in the bottom row we have the [[Lie groups]]

[[Spin(5)]] $\hookrightarrow$ [[Spin(6)]] $\hookrightarrow$ [[Spin(7)]] $\hookrightarrow$ [[Spin(8)]]

with their canonical [[subgroup]]-inclusions, while in the top row we have

[[SU(2)]] $\hookrightarrow$ [[SU(3)]] $\hookrightarrow$ [[G2]] $\hookrightarrow$ [[Spin(7)]]

and the right vertical inclusion $B \iota'$ is the [[delooping]] of one of the two non-standard inclusions, according to Prop. \ref{Spin7SubgroupsInSpin8}.

\end{prop}
\begin{proof}
The square on the right is that from Prop. \ref{G2IsIntersectionOfSpin7SubgroupsInSpin8}.

The square in the middle is \hyperlink{Varadarajan01}{Varadarajan 01, Lemma 9 on p. 10}.

The statement also follows with \hyperlink{Onishchik93}{Onishchik 93, Table 2, p. 144}:

$\backslash$begin\{center\} $\backslash$begin\{imagefromfile\} ``file\_name'': ``SO8SubgroupIntersection.jpg'', ``width'': 690 $\backslash$end\{imagefromfile\} $\backslash$end\{center\}

\end{proof}
\begin{prop}
\label{Spin3DotSpin5SubgroupsInSO8}\hypertarget{Spin3DotSpin5SubgroupsInSO8}{}
\textbf{([[Spin(5).Spin(3)]]-[[subgroups]] in [[SO(8)]])}

The [[direct product group]] [[SO(3)]] $\times$ [[SO(5)]] together with the groups [[Sp(2).Sp(1)]] and $Sp(1) \cdot Sp(2)$, with their canonical inclusions into [[SO(8)]], form 3 [[conjugacy classes of subgroups|conjugacy classes]] of [[subgroups]] inside [[SO(8)]], and the [[triality]] group $Out(Spin(8))$ [[action|acts]] [[transitive action|transitively]] on these three classes.

$\backslash$begin\{center\} $\backslash$begin\{xymatrix\} Sp(2)$\backslash$cdot Sp(1) $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}dr{\tt \symbol{94}}\{$\backslash$iota'\} $\backslash$ \& SO(8) \& SO(3) $\backslash$times SO(5) $\backslash$ar@\{\_\{(\}-{\tt \symbol{62}}\}l{\tt \symbol{94}}-\{$\backslash$iota\} $\backslash$ Sp(1) $\backslash$cdot Sp(2) $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}ur\emph{-\{ $\backslash$iota'` \} $\backslash$end\{xymatrix\} $\backslash$end\{center\}}

\end{prop}
(\hyperlink{Kollross02}{Kollross 02, Prop. 3.3 (3)})

Similarly:

\begin{prop}
\label{Spin3DotSpin5SubgroupsInSpin8}\hypertarget{Spin3DotSpin5SubgroupsInSpin8}{}
\textbf{([[Spin(5).Spin(3)]]-[[subgroups]] in [[Spin(8)]])}

The groups [[Spin(5).Spin(3)]], [[Sp(2).Sp(1)]] and $Sp(1) \cdot Sp(2)$, with their canonical inclusions into [[Spin(8)]], form 3 [[conjugacy classes of subgroups|conjugacy classes]] of [[subgroups]] inside [[Spin(8)]], and the [[triality]] group $Out(Spin(8))$ [[action|acts]] [[transitive action|transitively]] on these three classes.

$\backslash$begin\{center\} $\backslash$begin\{xymatrix\} Sp(2)$\backslash$cdot Sp(1) $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}dr{\tt \symbol{94}}\{$\backslash$iota'\} $\backslash$ \& Spin(8) \& Spin(3) $\backslash$cdot Spin(5) $\backslash$ar@\{\_\{(\}-{\tt \symbol{62}}\}l{\tt \symbol{94}}-\{$\backslash$iota\} $\backslash$ Sp(1) $\backslash$cdot Sp(2) $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}ur\emph{-\{ $\backslash$iota'` \} $\backslash$end\{xymatrix\} $\backslash$end\{center\}}

\end{prop}
(\hyperlink{CadekVanzura97}{Čadek-Vanžura 97, Sec. 2})

 In summary we have these subgroup inclusions

$\backslash$begin\{xymatrix\} $\backslash$mathrm\{Sp\}(1) \{$\backslash$cdot\} $\backslash$mathrm\{Sp\}(2) $\backslash$ar@\{=\}dddddr $\backslash$ar@\{\_\{(\}-{\tt \symbol{62}}\}ddrr|-\{ $\backslash$iota' \} $\backslash$ $\backslash$ \&\& $\backslash$mathrm\{Spin\}(8) $\backslash$ar@\{-\guillemotright{}\}dddddr \&\& $\backslash$mathrm\{Spin\}(3) \{$\backslash$cdot\} $\backslash$mathrm\{Spin\}(5) $\backslash$ar@\{-\guillemotright{}\}dddddr $\backslash$ar@\{\_\{(\}-{\tt \symbol{62}}\}ll|-\{$\backslash$iota\} $\backslash$ $\backslash$ $\backslash$mathrm\{Sp\}(2) \{$\backslash$cdot\} $\backslash$mathrm\{Sp\}(1) $\backslash$ar@\{=\}dddddr $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}uurr |\guillemotleft{}\guillemotleft{}\guillemotleft{}\guillemotleft{}\guillemotleft{}\guillemotleft{}\guillemotleft{}\guillemotleft{}\guillemotleft{}\guillemotleft{}\guillemotleft{}\guillemotleft{}\guillemotleft{}\{ $\backslash$phantom\{AA $\backslash$atop AA\} \} |\guillemotright{}\guillemotright{}\guillemotright{}\guillemotright{}\guillemotright{}\guillemotright{}\guillemotright{}\guillemotright{}\guillemotright{}\guillemotright{}\guillemotright{}{\tt \symbol{62}}\{ $\backslash$iota'` \} $\backslash$ \& $\backslash$mathrm\{Sp\}(1) \{$\backslash$cdot\} $\backslash$mathrm\{Sp\}(2) $\backslash$ar@\{\_\{(\}-{\tt \symbol{62}}\}ddrr|-\{ $\backslash$iota' \} $\backslash$ $\backslash$ \&\&\& $\backslash$mathrm\{SO\}(8) \&\& $\backslash$mathrm\{SO\}(3) $\backslash$times $\backslash$mathrm\{SO\}(5) $\backslash$ar@\{\_\{(\}-{\tt \symbol{62}}\}ll|-\{$\backslash$iota\} $\backslash$ $\backslash$ \& $\backslash$mathrm\{Sp\}(2) \{$\backslash$cdot\} $\backslash$mathrm\{Sp\}(1) $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}uurr|-\{ $\backslash$iota'` \} $\backslash$end\{xymatrix\}

permuted by triality:

$\backslash$begin\{center\} $\backslash$begin\{imagefromfile\} ``file\_name'': ``TrialityOnSp2Sp1.jpg'', ``width'': 680 $\backslash$end\{imagefromfile\} $\backslash$end\{center\}

\begin{quote}%
graphics grabbed from \hyperlink{FSS19}{FSS 19, Sec. 3.3}


\end{quote}
$\backslash$linebreak

\hypertarget{homotopy_groups}{}\subsubsection*{{Homotopy groups}}\label{homotopy_groups}

The [[homotopy groups]] of $SO(8)$ in [[low-dimensional topology|low degrees]]:

\begin{tabular}{l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l}
$G$&$\pi_1$&$\pi_2$&$\pi_3$&$\pi_4$&$\pi_5$&$\pi_6$&$\pi_7$&$\pi_8$&$\pi_9$&$\pi_10$&$\pi_11$&$\pi_12$&$\pi_13$&$\pi_14$&$\pi_15$\\
\hline 
$SO(8)$&$\mathbb{Z}_2$&$0$&$\mathbb{Z}$&$0$&$0$&$0$&$\mathbb{Z}^{\oplus 2}$&$\mathbb{Z}_{2}^{\oplus 3}$&$\mathbb{Z}_{2}^{\oplus 3}$&$\mathbb{Z}_{8} \oplus \mathbb{Z}_{24}$&$\mathbb{Z}_2 \oplus \mathbb{Z}$&0&$\mathbb{Z}^{\oplus 2}$&$\mathbb{Z}_2\oplus\mathbb{Z}_8\oplus\mathbb{Z}_{120}\oplus\mathbb{Z}_{2520}$&$\mathbb{Z}_2^{\oplus 7}$\\
\end{tabular}

$\backslash$linebreak

\hypertarget{cohomology_of_classifying_spaces}{}\subsubsection*{{Cohomology of classifying spaces}}\label{cohomology_of_classifying_spaces}

\begin{prop}
\label{CohomologyRingOfSpin8}\hypertarget{CohomologyRingOfSpin8}{}
The [[ordinary cohomology|ordinary]] [[cohomology ring]] of the [[classifying space]] $B Spin(8)$ is:

\textbf{1)} with [[coefficients]] in the [[cyclic group of order 2]]:

\begin{displaymath}
H^\bullet
  \big(
    B Spin(8), \mathbb{Z}_2
  \big)
  \;\simeq\;
  \mathbb{Z}
  \big[
    w_4, w_6, w_7, w_8, 
    \;
    \rho_2
    \left(
    \tfrac{1}{4}
    \left(
      p_2
      -
      \big(\tfrac{1}{2}p_1\big)^2
      -
      2 \chi
    \right)
    \right)
  \big]
\end{displaymath}
where $w_i$ are the [[universal characteristic class|universal]] [[Stiefel-Whitney classes]],

and where

\begin{displaymath}
\rho_2 \;\colon\; H^\bullet(B Spin(8), \mathbb{Z}) \to H^\bullet(B Spin(8), \mathbb{Z}_2)
\end{displaymath}
is [[cyclic group|mod 2 reduction]]

\textbf{2)} with [[coefficients]] in the [[integers]]:

\begin{displaymath}
H^\bullet
  \big(
    B Spin(8), \mathbb{Z}
  \big)
  \;\simeq\;
  \mathbb{Z}
  \Big[
    \tfrac{1}{2}p_1,
    \;
    \tfrac{1}{4}
    \left(
      p_2
      -
      \big(\tfrac{1}{2}p_1\big)^2
    \right)
    -
    \tfrac{1}{2}\chi
    ,
    \;
    \chi,
    \;
    \beta(w_6)
  \Big]
  /
  \big\langle 2 \beta(w_6)\big\rangle
  \,,
\end{displaymath}
where $p_1$ is the [[first fractional Pontryagin class]], $p_2$ is the [[second Pontryagin class]], $\chi$ is the [[Euler class]], and

\begin{displaymath}
\beta
  \;\colon\;
  H^\bullet
  \big(
    B Spin(8),
    \mathbb{Z}_2
  \big)
  \longrightarrow
  H^{\bullet + 1}
  \big(
    B Spin(8),
    \mathbb{Z}
  \big)
\end{displaymath}
is the [[Bockstein homomorphism]].

Moreover, we have the following relations:

\begin{displaymath}
\begin{aligned}
    \rho_2\left(
      \tfrac{1}{2}p_1
    \right)  
    & = 
    w_4
    \\
    \rho_2\big( \chi\big) 
    & =
    w_8
  \end{aligned}
\end{displaymath}
\end{prop}
This is due to \hyperlink{Quillen71}{Quillen 71}, \hyperlink{CadekVanzura95}{Čadek-Vanžura 95}, see \hyperlink{CadekVanzura97}{Čadek-Vanžura 97, Lemma 4.1}.

\begin{prop}
\label{}\hypertarget{}{}
Consider the [[looping and delooping|delooping]] of the [[triality]] automorphism relating [[Sp(2).Sp(1)]] with [[Spin(5).Spin(3)]] (Prop. \ref{Spin3DotSpin5SubgroupsInSpin8}) on [[classifying spaces]]

\begin{displaymath}
\itexarray{
    B 
    \big(
      Spin(5) \cdot Spin(3)
    \big)
    &\hookrightarrow&
    B Spin(8)
    \\
    \big\downarrow
    &&
    \big\downarrow^{ B \mathrlap{tri} }
    \\
    B 
    \big(
      Sp(2) \cdot Sp(1)
    \big)
    &\hookrightarrow&
    B Spin(8)    
  }
\end{displaymath}
Then the pullback of the [[universal characteristic classes]] of $B Spin(8)$ (from Prop. \ref{CohomologyRingOfSpin8}) along $B tri$ is as follows:

\begin{displaymath}
\big(
    B tri
  \big)^\ast
  \;\colon\;
  \begin{aligned}
    \tfrac{1}{2} p_1 & \mapsto \tfrac{1}{2} p_1
    \\
    \chi 
      & \mapsto 
      -
      \tfrac{1}{4}
      \big(
        p_2 
        - \big(\tfrac{1}{2}p_1\big)^2
      \big)
      + 
      \tfrac{1}{2}\chi
    \\
      \tfrac{1}{4}
      \big(
        p_2 
        - 
        \big(\tfrac{1}{2}p_1\big)^2
      \big)
      - 
      \tfrac{1}{2}\chi
    & \mapsto 
      -
      \chi
  \end{aligned}
\end{displaymath}
\end{prop}
(\hyperlink{CadekVanzura97}{Čadek-Vanžura 97, Lemma 4.2})

In fact $tri^{-1} = tri$.

Hence, in rational cohomology:

\begin{displaymath}
\begin{aligned}
    \big(
      B tri
    \big)^\ast
    \big(
      \tfrac{1}{4}p_2
    \big) 
    & =
    \big(
      B tri
    \big)^\ast
    \Big(
      \big(
        \tfrac{1}{4}p_2 
        - \big(\tfrac{1}{4}p_1\big)^2
        - \tfrac{1}{2}\chi
      \big)
      + \big(\tfrac{1}{4}p_1\big)^2
      + \tfrac{1}{2}\chi
    \Big) 
    \\
    & =
    -\chi
    + 
    \big(\tfrac{1}{4}p_1\big)^2
    -
    \tfrac{1}{2}
    \big(
      \tfrac{1}{4}p_2 
      - \big(\tfrac{1}{4}p_1\big)^2
      - \tfrac{1}{2}\chi
    \big)
  \end{aligned}
\end{displaymath}
$\backslash$linebreak

\hypertarget{structure_and_exceptional_geometry}{}\subsubsection*{{$G$-Structure and exceptional geometry}}\label{structure_and_exceptional_geometry}

[[!include Spin(8)-subgroups and reductions -- table]]

$\backslash$linebreak

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

[[!include low dimensional rotation groups -- table]]

$\backslash$linebreak

\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{itemize}%
\item [[Robert Bryant]], \emph{Remarks on Spinors in Low Dimension} (\href{https://services.math.duke.edu/~bryant/Spinors.pdf}{pdf}, [[BryantRemarksOnSpinorsInLowDimension.pdf:file]])

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{}

\end{itemize}
\hypertarget{subgroup_lattice_2}{}\subsubsection*{{Subgroup lattice}}\label{subgroup_lattice_2}

On the [[subgroup lattice]] of [[Spin(8)]]

\begin{itemize}%
\item A. L. Onishchik (ed.) \emph{Lie Groups and Lie Algebras}

\begin{itemize}%
\item \emph{I.} A. L. Onishchik, E. B. Vinberg, \emph{Foundations of Lie Theory},


\item \emph{II.} V. V. Gorbatsevich, A. L. Onishchik, \emph{Lie Transformation Groups}



\end{itemize}
Encyclopaedia of Mathematical Sciences, Volume 20, Springer 1993


\item [[Veeravalli Varadarajan]], \emph{Spin(7)-subgroups of SO(8) and Spin(8)}, Expositiones Mathematicae Volume 19, Issue 2, 2001, Pages 163-177 (, \href{https://core.ac.uk/download/pdf/81114499.pdf}{pdf})


\item [[Martin Čadek]], [[Jiří Vanžura]], \emph{On $Sp(2)$ and $Sp(2) \cdot Sp(1)$-structures in 8-dimensional vector bundles}, Publicacions Matemàtiques Vol. 41, No. 2 (1997), pp. 383-401 (\href{https://www.jstor.org/stable/43737249}{jstor:43737249})


\item Megan M. Kerr, \emph{New examples of homogeneous Einstein metrics}, Michigan Math. J. Volume 45, Issue 1 (1998), 115-134 (\href{https://projecteuclid.org/euclid.mmj/1030132086}{euclid:1030132086})


\item Andreas Kollross, Prop. 3.3 of \emph{A Classification of Hyperpolar and Cohomogeneity One Actions}, Transactions of the American Mathematical Society Vol. 354, No. 2 (Feb., 2002), pp. 571-612 (\href{https://www.jstor.org/stable/2693761}{jstor:2693761})



\end{itemize}
Discussion with an eye towards foundations of [[M-theory]]:

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation]]} (\href{https://arxiv.org/abs/1904.10207}{arXiv:1904.10207})

\end{itemize}
\hypertarget{cohomology}{}\subsubsection*{{Cohomology}}\label{cohomology}

The [[integral cohomology]] of the [[classifying spaces]] $B SO(8)$ and $B Spin(8)$ and the [[action]] of [[triality]] on these is discussed in

\begin{itemize}%
\item [[Alfred Gray]], Paul S. Green, \emph{Sphere transitive structures and the triality automorphism}, Pacific J. Math. Volume 34, Number 1 (1970), 83-96 (\href{https://projecteuclid.org/euclid.pjm/1102976640}{euclid:1102976640})


\item [[Daniel Quillen]], \emph{The mod 2 cohomology rings of extra-special 2-groups and the spinor groups}, Math. Ann . 194 (1971), 19


\item [[Martin Čadek]], [[Jiří Vanžura]], \emph{On the existence of 2-fields in 8-dimensional vector bundles over 8-complexes}, Commentationes Mathematicae Universitatis Carolinae, vol. 36 (1995), issue 2, pp. 377-394 (\href{https://dml.cz/handle/10338.dmlcz/118764}{dml-cz:118764})


\item [[Martin Čadek]], [[Jiří Vanžura]], Section 2 of \emph{On $Sp(2)$ and $Sp(2) \cdot Sp(1)$-structures in 8-dimensional vector bundles}, Publicacions Matemàtiques Vol. 41, No. 2 (1997), pp. 383-401 (\href{https://www.jstor.org/stable/43737249}{jstor:43737249})



\end{itemize}
[[!redirects Spin(8)]]

[[!redirects SO8]]

[[!redirects Spin8]]



\end{document}