Finite Observations and Recognition of the Compact Albert Algebra
Let J=H_3(O), the real algebra of 3×3 Hermitian matrices over the division octonions, with product a∘b=(ab+ba)/2 and diagonal trace T. A frame p=(p_0,p_1,p_2) consists of trace-one elements satisfying p_i∘p_i=p_i, p_i∘p_j=0 for i≠j, and p_0+p_1+p_2=1. For frames p,q,r, record the 27 pair traces T(p_i∘q_j), T(q_j∘r_k), T(r_k∘p_i), and 27 triple traces T((p_i∘q_j)∘r_k), for i,j,k∈{0,1,2}. How much do these traces determine? Each trace list in some open neighborhood of the data with pair entries 1/3 and triple entries 1/9, in a fixed twenty-dimensional chart, determines exactly two equivalence…
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