Appendix: Harmonic Curvature of a \(G_2\) Contact Structure
As already mentioned (see [2] for the general theory), the harmonic curvature of a \(G_2\) contact structure is a section of the bundle 
. This binary septic may be obtained by the Cartan equivalence method.
Specifically, a \(G_2\) contact structure on a five-dimensional manifold C may be specified in terms of an adapted co-frame as follows. Firstly, we choose a 1-form \(\omega ^0\) whose kernel is the contact distribution \(H\subset TC\). Non-degeneracy of the contact distribution says that \(\omega ^0\wedge d\omega ^0\wedge d\omega ^0\not =0\). The \(G_2\) structure is then determined by completing \(\omega ^0\) to a co-frame \(\omega ^0,\omega ^1,\omega ^2,\omega ^3,\omega ^4\) so that
$$\begin{aligned} d\omega ^0=\chi (\omega ^1\wedge \omega ^4-3\omega ^2\wedge \omega ^3) \bmod \omega ^0 \end{aligned}$$
(27)
for some smooth function \(\chi \). Specifically, if \(X_0,X_1,X_2,X_3,X_4\) is the dual frame, then \(H={\mathrm {span}}\{X_1,X_2,X_3,X_4\}\) and the twisted cubic (21) may be given as
$$\begin{aligned} (s,t)\mapsto s^3X_1+s^2tX_2+st^2X_3+t^3X_4, \end{aligned}$$
(28)
compatibility with the Levi form being a consequence of (27). Imposing the structure equations (27) leaves precisely the following freedom in choice of co-frame:
$$\begin{aligned} \left[ \begin{array}{c}{\tilde{\omega }}^0\\ {\tilde{\omega }}^1\\ {\tilde{\omega }}^2\\ {\tilde{\omega }}^3\\ {\tilde{\omega }}^4\end{array}\right] =\left[ \begin{array}{ccccc} t_9&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ t_{10}&{}\quad t_5{}^3&{}\quad 3t_5{}^2t_6&{}\quad 3t_5t_6{}^2&{}\quad t_6{}^3\\ t_{11}&{}\quad t_5{}^2t_7&{}\quad t_5(t_5t_8+2t_6t_7)&{}\quad t_6(2t_5t_8+t_6t_7)&{}\quad t_6{}^2t_8\\ t_{12}&{}\quad t_5t_7{}^2&{}\quad t_7(2t_5t_8+t_6t_7)&{}\quad t_8(t_5t_8+2t_6t_7)&{}\quad t_6t_8{}^2\\ t_{13}&{}\quad t_7{}^3&{}\quad 3t_7{}^2t_8&{}\quad 3t_7t_8{}^2&{}\quad t_8{}^3 \end{array}\right] \, \left[ \begin{array}{c}\omega ^0\\ \omega ^1\\ \omega ^2\\ \omega ^3\\ \omega ^4\end{array}\right] \end{aligned}$$
(29)
for arbitrary functions \(t_5,t_6,t_7,t_8,t_9,t_{10},t_{11},t_{12},t_{13}\) on C subject only to \(t_9(t_5t_8-t_6t_7)\not =0\). (The functions \(t_5,t_6,t_7,t_8\) correspond to
$$\begin{aligned} {(s,t)\mapsto (s,t)} \left( \begin{array}{cc}t_5&{}\quad t_7\\ t_6&{}\quad t_8\end{array}\right) \end{aligned}$$
as a change of parameterisation in (28) whilst \(t_9,t_{10},t_{11},t_{12},t_{13}\) modify the co-frame with multiples of \(\omega ^0\).) To proceed with Cartan’s method of equivalence, we now pass to the bundle \(\widetilde{C}\rightarrow C\) whose sections are frames adapted according to (27). It is a G-principal bundle where G is the 9-dimensional Lie subgroup of \({\mathrm {GL}}(5,{\mathbb {R}})\) given in (29) and comes tautologically equipped with 1-forms \(\theta ^0,\theta ^1,\theta ^2,\theta ^3,\theta ^4\) whose pull-backs along a section are \(\omega ^0,\omega ^1,\omega ^2,\omega ^3,\omega ^4\), the co-frame on C corresponding to that section. Cartan’s aim is to extend this to an invariant co-frame on \(\widetilde{C}\) by making various normalisations. For our purposes we need not take these normalisations too far. For calculation, we choose a co-frame on C adapted according to (27) so that \(\widetilde{C}\) is identified as \(G\times C\) and the forms \(\theta ^0,\theta ^1\,\theta ^2,\theta ^3,\theta ^4\) are given as
$$\begin{aligned} \left[ \begin{array}{c}\theta ^0\\ \theta ^1\\ \theta ^2\\ \theta ^3\\ \theta ^4\end{array}\right] =\left[ \begin{array}{ccccc} t_9&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ t_{10}&{}\quad t_5{}^3&{}\quad 3t_5{}^2t_6&{}\quad 3t_5t_6{}^2&{}\quad t_6{}^3\\ t_{11}&{}\quad t_5{}^2t_7&{}\quad t_5(t_5t_8+2t_6t_7)&{}\quad t_6(2t_5t_8+t_6t_7)&{}\quad t_6{}^2t_8\\ t_{12}&{}\quad t_5t_7{}^2&{}\quad t_7(2t_5t_8+t_6t_7)&{}\quad t_8(t_5t_8+2t_6t_7)&{}\quad t_6t_8{}^2\\ t_{13}&{}\quad t_7{}^3&{}\quad 3t_7{}^2t_8&{}\quad 3t_7t_8{}^2&{}\quad t_8{}^3 \end{array}\right] \, \left[ \begin{array}{c}\omega ^0\\ \omega ^1\\ \omega ^2\\ \omega ^3\\ \omega ^4\end{array}\right] . \end{aligned}$$
Step 0 Normalise the co-frame on C so that
$$\begin{aligned} d\omega ^0=\chi (\omega ^1\wedge \omega ^4-3\omega ^2\wedge \omega ^3). \end{aligned}$$
(30)
This is easily achieved by the freedom
$$\begin{aligned} {\tilde{\omega }}^1=\omega ^1+t_{10}\omega ^0,{\tilde{\omega }}^2=\omega ^1+t_{11}\omega ^0,{\tilde{\omega }}^3=\omega ^1+t_{12}\omega ^0,{\tilde{\omega }}^4=\omega ^1+t_{13}\omega ^0 \end{aligned}$$
and determines the functions \(t_{10},t_{11},t_{12},t_{13}\).
Step 1 Find \(\theta ^5\) such that
$$\begin{aligned} d\theta ^0 =-6\theta ^0\wedge \theta ^5+\theta ^1\wedge \theta ^4-3\theta ^2\wedge \theta ^3. \end{aligned}$$
This may be achieved by setting \(-t_9=\Delta ^3/\chi \), where \(\Delta =t_6t_7-t_5t_8\) and then
$$\begin{aligned} \theta ^5=-\frac{1}{6}\frac{d\chi }{\chi }+\frac{1}{2}\frac{d\Delta }{\Delta } +\frac{\chi }{\Delta ^3}\Big (\frac{t_{13}\theta ^1}{6}-\frac{t_{12}\theta ^2}{2} +\frac{t_{11}\theta ^3}{2}-\frac{t_{10}\theta ^4}{6}\Big )+s\theta ^0 \end{aligned}$$
for some function s.
Step 2 Find \(\theta ^7,\theta ^8,\theta ^9\) such that
$$\begin{aligned} E^1\equiv d\theta ^1-(6\theta ^0\wedge \theta ^9-3\theta ^1\wedge \theta ^5 -3\theta ^1\wedge \theta ^8+3\theta ^2\wedge \theta ^7) \end{aligned}$$
is of the form \(c^1{}_{\mu \nu }\theta ^\mu \wedge \theta ^v\) for \(\mu ,\nu =0,1,2,3,4\). It follows that
$$\begin{aligned} E^1\wedge \theta ^0\wedge \theta ^1\wedge \theta ^2 =c^1{}_{34}\theta ^0\wedge \theta ^1\wedge \theta ^2\wedge \theta ^4\wedge \theta ^5. \end{aligned}$$
It turns out that
$$\begin{aligned} c^1{}_{34}=\frac{t_5{}^7}{\Delta ^5} \Big (\psi _0+\psi _1s+\psi _2s^2+\psi _3s^3+\psi _4s^4+\psi _5s^5+\psi _6s^6 +\psi _7s^7\Big ), \end{aligned}$$
where \(\psi _0,\psi _1,\psi _2,\psi _3,\psi _4,\psi _5,\psi _6,\psi _7\) are functions on C. These are the coefficients of the invariantly defined harmonic curvature. In fact, if
$$\begin{aligned} d\omega ^0=\chi (\omega ^1\wedge \omega ^4-3\omega ^2\wedge \omega ^3) \end{aligned}$$
on C and we write
$$\begin{aligned} d\omega ^i=\sum _{1\le j<k\le 4}a^i{}_{jk}\omega ^j\wedge \omega ^k \bmod \omega ^0,\text{ for } i=1,2,3,4 \end{aligned}$$
(31)
then
$$\begin{aligned} \begin{array}{rcl}\psi _0&{}=&{}a^1{}_{34}\\ \psi _1&{}=&{}-2a^1{}_{24}+3a^2{}_{34}\\ \psi _2&{}=&{}3a^1{}_{14}+a^1{}_{23}-6a^2{}_{24}+3a{}^3{}_{34}\\ \psi _3&{}=&{}-2a^1{}_{13}+9a^2{}_{14}+3a^2{}_{23}-6a^3{}_{24}+a^4{}_{34}\\ \psi _4&{}=&{}a^1{}_{12}-6a^2{}_{13}+9a^3{}_{14}+3a^3{}_{23}-2a^4{}_{24}\\ \psi _5&{}=&{}3a^2{}_{12}-6a^3{}_{13}+3a^4{}_{14}+a^4{}_{23}\\ \psi _6&{}=&{}3a^3{}_{12}-2a^4{}_{13}\\ \psi _7&{}=&{}a^4{}_{12}. \end{array} \end{aligned}$$
(32)
From the general theory of parabolic geometry [2, Theorem 3.1.12] we find the following characterisation of flat \(G_2\) contact structures.
Theorem 5
The local symmetry algebra of the \(G_2\) contact structure specified by a co-frame adapted according to (30) is the split exceptional Lie algebra \(G_2\) if and only if \(\psi _0,\psi _1,\psi _2,\psi _3,\psi _4,\psi _5,\psi _6,\psi _7\) given by (32) all vanish. Moreover, in this case, the manifold C is locally isomorphic to the homogeneous model.
Remark
The formulæ (31) and (32) may alternatively be derived as follows. According to (20), the exterior derivative \(d:\wedge ^1\rightarrow \wedge ^2\) gives rise, via the diagram
to an invariantly defined first order differential operator
In fact, this is nothing more than the second operator in the Rumin complex [16], which depends only on the contact structure on C. The exterior derivative, on the other hand, may be written as \(\omega _b\mapsto \nabla _{[a}\omega _{b]}\) for any torsion-free connection \(\nabla _a\) on TC. Thus, the Rumin operator (33) may be written with spinor indices [15], adapted to our cause, as
$$\begin{aligned} \omega _{ABC}\longmapsto \nabla _{(AB}{}^H\omega _{CD)H}. \end{aligned}$$
(34)
One may readily check that the formulæ (31) and (32) amount to the stipulation that
$$\begin{aligned} \psi _{ABCDEFG}\pi ^A\pi ^B\pi ^C\pi ^D\pi ^E\pi ^F\pi ^G =\pi ^A\pi ^B\pi ^C\pi ^D\nabla _{AB}{}^H(\pi _C\pi _D\pi _H) \end{aligned}$$
(35)
for all sections \(\pi _A\) of 
. Note, by the Leibniz rule
$$\begin{aligned} \begin{array}{rcl} \pi ^A\pi ^B\pi ^C\pi ^D\nabla _{AB}{}^H(f\pi _C\pi _D\pi _H) &{}=&{}f\pi ^A\pi ^B\pi ^C\pi ^D\nabla _{AB}{}^H(\pi _C\pi _D\pi _H)\\ &{}&{}{}+\pi ^A\pi ^B\underbrace{\pi ^C\pi ^D\pi _C}_{=0}\pi _D\pi _H \nabla _{AB}{}^Hf,\end{array} \end{aligned}$$
that the right hand side of (35) is homogeneous of degree 7 over the functions and, therefore, automatically of the form given on the left. It follows that \(\psi _{ABCDEFG}\) is the obstruction to writing (33) as
$$\begin{aligned} \omega _{ABC}\longmapsto {\mathcal {D}}_{(AB}{}^H\omega _{CD)H}, \end{aligned}$$
where \({\mathcal {D}}_{ABC}\) is induced by a connection on 
, because the Leibniz rule for such a connection would imply that
$$\begin{aligned} {\mathcal {D}}_{AB}{}^H(\pi _C\pi _D\pi _H) =\pi _C\pi _D{\mathcal {D}}_{AB}{}^H\pi _H +\pi _C\pi _H{\mathcal {D}}_{AB}{}^H\pi _D +\pi _D\pi _H{\mathcal {D}}_{AB}{}^H\pi _C \end{aligned}$$
and the right hand side of (35) would therefore vanish. In other words, the formula (34) depends on \(\nabla _a\) being torsion-free and \(\psi _{ABCDEFG}\) may be seen as some invariant part of the partial torsion of a freely chosen spinor connection \({\mathcal {D}}_a:S\rightarrow \wedge ^1\otimes S\).
More specifically, suppose \({\mathcal {D}}_a:S\rightarrow \wedge ^1\otimes S\) is any connection and define its partial torsion\(T_{ABCD}{}^{EFG}=T_{(ABCD)}{}^{(EFG)}\) according to
$$\begin{aligned} {\mathcal {D}}_{(AB}{}^E{\mathcal {D}}_{CD)E}f =T_{ABCD}{}^{EFG}{\mathcal {D}}_{EFG}f,\forall \text{ smooth } \text{ functions } f. \end{aligned}$$
Changing the connection \({\mathcal {D}}_a\), leads to a change of partial connection
according to
$$\begin{aligned} \widehat{\mathcal {D}}_{ABC}\pi _D ={\mathcal {D}}_{ABC}\pi _D-\Gamma _{ABCD}{}^E\pi _E, \text{ where } \Gamma _{ABCD}{}^E=\Gamma _{(ABC)D}{}^E \end{aligned}$$
and, therefore, an induced change on 
, namely
$$\begin{aligned} \widehat{\mathcal {D}}_{ABC}\omega _{DEF}= {\mathcal {D}}_{ABC}\omega _{DEF}-3\Gamma _{ABC(D}{}^G\omega _{EF)G}. \end{aligned}$$
It follows that
$$\begin{aligned} \widehat{\mathcal {D}}_{AB}{}^E\widehat{\mathcal {D}}_{CDE}f ={\mathcal {D}}_{AB}{}^E{\mathcal {D}}_{CDE}f -3\Gamma _{AB}{}^E{}_{(C}{}^G{\mathcal {D}}_{DE)G}f \end{aligned}$$
and, therefore, that
$$\begin{aligned} \widehat{\mathcal {D}}_{(AB}{}^E\widehat{\mathcal {D}}_{CD)E}f = {\mathcal {D}}_{(AB}{}^E{\mathcal {D}}_{CD)E}f -2\Gamma _{(AB}{}^E{}_C{}^F{\mathcal {D}}_{D)EF}f +\Gamma _{H(AB}{}^{HE}{\mathcal {D}}_{CD)E}f \end{aligned}$$
whence the partial torsion of \({\mathcal {D}}_a\) changes according to
$$\begin{aligned} \widehat{T}_{ABCD}{}^{EFG}=T_{ABCD}{}^{EFG} -2\Gamma _{(AB}{}^{(E}{}_C{}^F\delta _{D)}{}^{G)} +\Gamma _{H(AB}{}^{H(E}\delta _C{}^F\delta _{D)}{}^{G)}. \end{aligned}$$
In particular, the trace-free part of \(T_{ABCD}{}^{EFG}\), equivalently
$$\begin{aligned} \psi _{ABCDEFG}\equiv T_{(ABCDEFG)}, \end{aligned}$$
is an invariant of the \(G_2\) contact structure.