Let (V, g, D) be an oriented Lorentz vector space, and let ε = (e1, …, em) be an orthonormal basis for V that is positively oriented with respect to D. We denote by Lm the set of linear isometries on V:
where we recall that Lin(V, V) is the vector space of linear maps from V to V.
Let A be a map in Lin(V, V). For brevity, we denote
by , and likewise omit &ip.eop; from the notation for other matrices. Corresponding to (4.4.1) and (4.4.2), we have
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