Taking the first spatial direction as compactification direction is convenient, for it does not change the conventions on the simple roots. More precisely, the Kaluza–Klein ansatz is compatible in that case with our Iwasawa decomposition (2.8View Equation) of the spatial metric with 𝒩 an upper triangular matrix. The (equivalent) choice of the tenth direction as compactification direction would correspond to a different (equivalent) choice of 𝒩.