Sums of squares, the octonions, and (7,3,1) - MAA Sections

O is another name for (7, 3, 1)
Seven quaternion subalgebras of O
O contains seven complex subalgebras C n = R〈e n 〉 and seven
quaternion subalgebras H n = R〈e t , e u , e v 〉, where {t, u, v} is a block
in (7, 3, 1)
Each H n contains three of the C k .
Each C k is contained in three of the H n .
Each pair {C k , C m } is contained in a unique H n together.
The (7, 3, 1) design sits inside O
The above design of subalgebras of O has the same incidence matrix
as (7, 3, 1).
The two designs are the same!
Brown
The Octonions