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The general non-orthogonal Cayley-Dickson construction

In [3, §4], Garibaldi-Petersson have introduced a special version of the non-orthogonal Cayley-Dickson construction that uses a purely inseparable field extension K/k of characteristic 2 and exponent at most 1 together with a k-linear form s : K → k taking 1 into 1 and an arbitrary scalar μ in the base field to exhibit a flexible conic algebra Cay(K;μ, s) having k-dimension twice the degree of K/k. In the present set of notes we intend to investigate the non-orthogonal Cayley-Dickson construction in its most general form. In doing so, we are no longer confined to base fields of characteristic 2 but will, in fact, be able to work instead over an arbitrary commutative ring, again denoted by k.