= Global frame on four-dimensional de Sitter spacetime
{title2=$E_0=\partial_\tau,\quad E_i=(\cosh\tau)^{-1}Y_i$}
In global coordinates the <de Sitter spacetime> metric is $-d\tau^2+\cosh^2\tau\,g_{S^3}$. A global <frame of a vector bundle> orthonormal for the round metric $Y_1,Y_2,Y_3$ on $S^3$ gives the displayed global <pseudo-orthonormal frame>. For example, identify $S^3$ with the <unit quaternions> and use $qi,qj,qk$. Choosing the spatial orientation and the future direction consistently gives a <section of a fiber bundle> for the oriented, time-oriented frame bundle. The <principal bundle trivialization by a global section> then proves that bundle is trivial.
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