Basis vectors, and bivector imaginaries in 3-space

σx = {0, {1, 0, 0}, {0, 0, 0}, 0} ; σy = {0, {0, 1, 0}, {0, 0, 0}, 0} ; σz = {0, {0, 0, 1}, {0, 0, 0}, 0} ; (* unit vectors *)

Mul3[σx, Mul3[σy, σz]]//MatrixForm (* the product of all three is the trivector imaginary *)

( {{0}, {{0, 0, 0}}, {{0, 0, 0}}, {1}} )

σix = Mul3[σy, σz] ; σiy = Mul3[σz, σx] ; σiz = Mul3[σx, σy] ; (* unit bivectors *)

σix//MatrixForm

( {{0}, {{0, 0, 0}}, {{1, 0, 0}}, {0}} )

Mul3[σix, σix]//MatrixForm (* these a bivector imaginaries *)

( {{-1}, {{0, 0, 0}}, {{0, 0, 0}}, {0}} )

Mul3[σiy, σiy]//MatrixForm

( {{-1}, {{0, 0, 0}}, {{0, 0, 0}}, {0}} )

Mul3[σiz, σiz]//MatrixForm

( {{-1}, {{0, 0, 0}}, {{0, 0, 0}}, {0}} )

Mul3[σiy, σiz]//MatrixForm (* sequential basis bivectors multiply to give minus the third ... *)

( {{0}, {{0, 0, 0}}, {{-1, 0, 0}}, {0}} )

Mul3[σix, Mul3[σiy, σiz]]//MatrixForm (* ... so the product of three unit bivectors is 1 *)

( {{1}, {{0, 0, 0}}, {{0, 0, 0}}, {0}} )

Mul3[2 + σx - 2σiy + 1.5 i3, -3 + σx - 3σy + 4σiz]//MatrixForm (* a generic product *)

( {{-15.5}, {{-16.5, -0.5, -0.5}}, {{-28., -20., -28.5}}, {-39.5}} )

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