Using the vielbein, we can convert between base-space and tangent-space coordinates. In particular, by writing the four-form eld strength in equation (5.1c) in tangent-space coordinates, one nds that
FABCD= 6k
R2ABCD, (E.8)
where 0123 = 1 and all other non-zero components are related by antisymmetry. Furthermore, if one takes the exterior derivative of equation (E.5), plugs this into equation (5.1d), and converts to tangent-space coordinates, one nds that
FAB= 2k
R2AB, (E.9)
where45=67=89= 1and all other non-zero components are related by antisymmetry. Equations (5.1b), (E.8), and (E.9) then imply that
eφΓ·F2 = 2
R Γ45+ Γ67+ Γ89 , eφΓ·F4 = 6
RΓ0123. Plugging these expressions into equation (5.12) then gives
Γ·F = 1 4R
−Γ11 Γ45+ Γ67+ Γ89
+ 3Γ0123
. (E.10)
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