An Exceptional 7-dimensional Real Algebra: Octonions, $G_2$, and the Fano Plane
We study a seven-dimensional non-associative algebra, the \emph{exceptional Vidinli algebra}, defined by lifting the bilinear product introduced by H\"{u}seyin Tevfik Pasha (Vidinli) in 1882 from three to seven dimensions via the octonionic cross product. This algebra is unital, simple, and non-associative, with automorphism group $U(3)$. Its multiplication splits canonically into a simple Jordan algebra and a Heisenberg Lie algebra, realizing the Jordan--Lie structure of the exceptional Vidinli algebra. Every principal 2-plane through the unit is isomorphic to $\C$, and every principal 3-plane is isomorphic to a twisted Vidinli algebra introduced below. %as the parameter $t$ varies, the complete twisted family $\{\mathcal{V}_t\}$ is realized inside $\V_7$. The main result is a $(\Z/2)^3$ grading of the cross product, under which the multiplication table of the exceptional Vidinli algebra is determined by three explicit rules with no reference to the calibration form. This grading unifies three structures through a single group: the Fano plane $\mathrm{PG}(2,2)$, whose lines correspond to the Vidinli subalgebras and Vidinli-Jordan subalgebras, %of $\V_7$, the discrete family of directional Vidinli algebras indexed by the nonzero elements of $(\Z/2)^3$, and the Heisenberg partition of basis pairs, which realizes the Fano incidence relation as a commutator condition. The resulting Fano-Vidinli duality identifies $(\Z/2)^3$ as the common source of both the Fano geometry and the Vidinli family.