Reversion, and physical components formerly known as imaginary

Mul2[avec, bvec]//MatrixForm (* here the bivector part of the geometric product ab is the physical component formerly known as imaginary *)

( {{ax bx + ay by}, {{0, 0}}, {-ay bx + ax by}} )

Mul2[bvec, avec]//MatrixForm

( {{ax bx + ay by}, {{0, 0}}, {ay bx - ax by}} )

Simplify[Mul2[Mul2[avec, bvec], Mul2[bvec, avec]]]//MatrixForm

( {{(ax^2 + ay^2) (bx^2 + by^2)}, {{0, 0}}, {0}} )

Rev2[A_] := {A[[1]], {A[[2, 1]], A[[2, 2]]}, -A[[3]]}

Simplify[Mul2[Mul2[avec, bvec], Rev2[Mul2[avec, bvec]]]]//MatrixForm

( {{(ax^2 + ay^2) (bx^2 + by^2)}, {{0, 0}}, {0}} )

Rev2[avec]

{0, {ax, ay}, 0}

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