A Jacobson Radical Decomposition of the Fano-Snowflake Configuration

The Fano-Snowflake, a specific $non$-unimodular projective lattice configuration associated with the smallest ring of ternions $R_{\mathrm{♢}}$ (arXiv:0803.4436 and 0806.3153), admits an interesting partitioning with respect to the Jacobson radical of $R_{\mathrm{♢}}$. The totality of 21 free cyclic submodules generated by non-unimodular vectors of the free left $R_{\mathrm{♢}}$-module $R_{\mathrm{♢}}^3$ are shown to split into three disjoint sets of cardinalities 9, 9 and 3 according as the number of Jacobson radical entries in the generating vector is 2, 1 or 0, respectively. The corresponding “ternion-induced” factorization of the lines of the Fano plane sitting in the middle of the Fano-Snowflake (6 – 7 – 3) is found to $differ fundamentally$ from the natural one, i. e., from that with respect to the Jacobson radical of the Galois field of two elements (3 – 3 – 1).