Let I be the exceptional 27-dimensional Jordan algebra over C . Its automorphism group is the Lie group F 4 C and this group is known to have a finite subgroup AL, where A is a self centralizing elementary abelian of order 27, L≅ SL(3, 3), and L normalizes A. As an A-module, I decomposes into a direct sum of 1-dimensional spaces I x which afford the 27 distinct linear characters xϵA^:=Hom( A, C x). These spaces satisfy I x I y = I xy . Let ω = e 2πi 3 . There are a basis of I of the form e x , for xϵA^, and a function g: A^× A^→ F 3 such that (∗) e x e y = (−2) e( x, y) ω g( x, y) e xy , where c( x, y) = 0 if x and y are linearly dependent and c( x, y) = 1 otherwise. Identifying A^ with F 3 3, we write x = ( x 1, x 2, x 3) and y = ( y 1, y 2, y 3). A function g which has the above properties is g( x, y) = − x 1, x 2, x 3 − x 3, y 1, y 2 + x 2, x 3, y 1 + x 1, y 2, y 3. The elements e x | x ϵ A^ generate the infinite commutative loop L :={(−2) m ω n e x | mϵ Z , nϵ Z 3, xϵA^} under Jordan multiplication. The loop S is not Moufang but has as quotient a Moufang loop M of order 81 and exponent 3. Conversely, the loop M may be constructed from scratch (using g) and used to define the Jordan algebra I using the formula (∗); this gives a new existence proof for a simple 27-dimensional Jordan algebra over fields of characteristic not 2 or 3 with a primitive cube root of unity (in characteristic 3, we get the group algebra of A^). We discuss some finite groups associated to M and the Lie groups F 4( C ) and 3 E 6( C ) and compare the analogous situation with the loop O 16, the Cayley numbers, and Lie groups G 2( C ) and D 4( C ). We also get a new construction of the cubic form in 27 variables whose group is 3 E 6( C ) and an easy and natural construction of the exotic 3-local subgroup 3 1+3+3: SL(3, 3).