#hypercomplex numbers

The fano plane in Octonian algebra - hypercomplex numbers [8D]

Octionions are non-commutative and non-associative, say, the order of calculating is important.

x·y ≠ y·x (non-commutative),

x·(y·z) ≠ (x·y)·z (non-associative)

For multiplication in the fano plane, you take two neighbouring elements on a line and the resulting value is the third element that closes the loop. Moving with arrows gives a positive value, moving against arrows gives a negative value.

Interestingly the symmetry on those values is still there, only inverted. As example:

e_3·e_4=e_6 and e_4·e_3= (-e_6)

Hence:

e_n·e_m=e_l and e_m·e_n= (-e_l)