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822 RUGGERO MARIA SANTILLIZ = Diag(1, 1, 1, i/n), Z × Z † ≠ I. (6.1.4b)The use of rotations and Lorentz transforms then yields a lifting of all remainingcomponents of the isometric. The Lie-Santilli isotheory is constructed by applying,for reasons clarified below, the inverse of the metric transform to the totalityof the mathematics underlying Lie’s theory, resulting in expressions of the typeU × U † = (Z × Z † ) −1 = Diag.(1/b 2 1, 1/b 2 2, 1/b 2 3, 1/b 2 4), == Diag.(n 2 1, n 2 2, n 2 3, n 2 4) (6.1.5a)I → Î = U × I × U † = Diag.(1/b 2 1, 1/b 2 2, 1/b 2 3, 1/b 2 4) = Diag.(n 2 1, n 2 2, n 2 3, n 2 4),n α = n α (µ, τ, ω, ...), n 4 = n,n ∈ R → ˆn = U × n × U † = n × (U × U † ) = n × Î ∈ ˆR,(6.1.5b)(6, 1.5c)n × m → ˆn ˆ× ˆm = U × (n × m) × U † = ˆn × ˆT × ˆm, ˆT = 1/U × U † , (6, 1.5d)[X i , X j ] = X i × X j − X j × X i → [ ˆX iˆ, ˆX j ] = ˆX i ˆ× ˆX j − ˆX j ˆ× ˆX i = U × [X i , X j ] × U † ,(6.1.5e)e X → ê ˆX = U × (e X ) × U † = (e X× ˆT ) × Î = Î × (e ˆT ×X ), etc.(6.1.5f)The invariance under additional nonunitary transforms is assured, providedthat it is studied within the context of isomathematics and not that of conventionalmathematics. This requires the identical reformulation of a given nonunitarytransform into the isounitary transform,W × W † ≠ I, W = Ŵ × ˆT 1/2 , W × W † ≡ Ŵ ˆ×Ŵ ˆ† = Ŵ ˆ† × Ŵ = Î, (6.1.6)under which we have the invariance lawsÎ → Ŵ ˆ×Î ˆ×Ŵ ˆ† ≡ Î,(6.1.7a)ˆX i ˆ× ˆX j → Ŵ ˆ×( ˆX i ˆ× ˆX j ) ˆ×Ŵ ˆ† = ˆX ′ i × ˆT × ˆX ′ j = ˆX ′ i ˆ× ˆX ′ j, etc. (6.1.7b)from which all other invariances follow. Note the invariance of the numericalvalue of the isounit Î and of the isoproduct represented by the numerical invarianceof ˆT .The application of the above formalism to the invariance of locally varyingspeeds of light was achieved for the first time by R. M. Santilli in paper [4a] of1983 at the classical level and in paper [4b] of the same year for the operatorcounterpart, to be studied in detail in subsequent papers [5] and additional ones.These studies achieved the invariance of the following universal isoline isoelementon the Minkowski-Santilli isospace ˆM(ˆx, ˆη, ˆR)ˆxˆ2 = ˆx µ ˆ×ˆη µν ˆ×ˆx ν = [x µ × ˆη µν (x, d, τ, ω, ...) × x ν ] × Î =