  The Fano-Snowflake, a specific $non$-unimodular projective lattice
configuration associated with the smallest ring of ternions $R_{\diamondsuit}$
(arXiv:0803.4436 and 0806.3153), admits an interesting partitioning with
respect to the Jacobson radical of $R_{\diamondsuit}$. The totality of 21 free
cyclic submodules generated by non-unimodular vectors of the free left
$R_{\diamondsuit}$-module $R_{\diamondsuit}^{3}$ 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).
