This bridge note presents a real two-dimensional matrix realization of the imaginary unit within a unital real-algebra framework. Let V = ℝ² and let J in EndR (V) be the standard linear complex structure satisfying J² = −I. The two-dimensional unital commutative real subalgebra AJ = xI + yJ: x, y in ℝ ⊂ EndR (V) is shown to be isomorphic to ℂ through the map Φ (x+iy) = xI + yJ, with Φ (i) = J. The note then formulates Euler’s identity in real matrix form: Φ (e^ (iθ) ) = exp (θJ) = I cos θ + J sin θ. It then introduces a Lorentzian companion construction. With the Lorentzian metric η = diag (1, −1), the companion operator K = Jη satisfies K² = I, and its exponential generates the standard 1+1-dimensional Lorentz boost. In a null basis, the boost diagonalizes as diag (e^δ, e^ (−δ) ). The document is intended as a compact bridge note for Arc Geometry applications. Its formal mathematical core is separated from the arc-geometric interpretive layer, where projection, time-torque, and holographic-track language are treated as research-development vocabulary rather than as additional assumptions in the proofs.
Frank F. (Arcman) Meng (Fri,) studied this question.