Lemma. Finite hyperbolic reduction [chevalley1954algebraic, II.2.9, pp. 65--66] [fcap-0014]
Lemma. Finite hyperbolic reduction [chevalley1954algebraic, II.2.9, pp. 65--66] [fcap-0014]
Chevalley splits a finite-dimensional real quadratic space into hyperbolic planes and a definite remainder. TauCeti combines that classical reduction with iteration of Theorem [fcap-0013] and Kronecker equivalences to synthesize the explicit tensor and matrix packaging below. Iterating \(n\) times gives \[\mathcal {C}\kern -2pt\ell _{p+n,q+n}\simeq \mathcal {C}\kern -2pt\ell _{p,q}\otimes _{\mathbb {R}}M_{2^n}(\mathbb {R}).\] Taking \(n=\min (p,q)\) removes the common positive and negative part: \[\mathcal {C}\kern -2pt\ell _{p,q}\simeq \mathcal {C}\kern -2pt\ell _{p-n,q-n}\otimes _{\mathbb {R}}M_{2^n}(\mathbb {R}), \qquad n=\min (p,q).\] Thus if \(p\le q\), the residual algebra is \(\mathcal {C}\kern -2pt\ell _{0,q-p}\); if \(q\le p\), it is \(\mathcal {C}\kern -2pt\ell _{p-q,0}\). The matrix factors combine through the Kronecker equivalence \[M_{2^a}(\mathbb {R})\otimes M_{2^b}(\mathbb {R}) \simeq M_{2^{a+b}}(\mathbb {R}).\] Chevalley supplies the quadratic-space decomposition. The displayed \(M_{2^n}\) packaging and its named equivalences are TauCeti's formalized synthesis.