So octonion set provides the largest normed division algebra, and starting with sedenions, Cayley-Dickson construction provides algebras with zero divisors.

From what I understand, it means there are pairs of non-null sedenion $(s_a,s_b)$ for which $ s_a·s_b = s_0$ is correct – where $s_0$ stands for the null sedenion $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$. But maybe I already misunderstood something here?

But given that higher order algebra gradually lose algebraic properties as the scale of dimension is climbed, are they any pair of non-null sedenion $(s_c, s_d)$ for which $s_c/s_0=s_d$ holds? That is, is it valid, at least in some cases, to divide by the null sedenion?