Quaternionic root systems and subgroups of the $Aut(F4)$

Cayley-Dickson doubling procedure is used to construct the root systems of some celebrated Lie algebras in terms of the integer elements of the division algebras of real numbers, complex numbers, quaternions, and octonions. Starting with the roots and weights of $SU(2)$ expressed as the real numbers one can construct the root systems of the Lie algebras of $SO(4),SP(2)≈SO(5),SO(8),SO(9),F4$ and $E8$ in terms of the discrete elements of the division algebras. The roots themselves display the groups structures besides the octonionic roots of $E8$ which form a closed octonion algebra. The automorphism group $Aut(F4)$ of the Dynkin diagram of $F4$ of order 2304, the largest crystallographic group in four-dimensional Euclidean space, is realized as the direct product of two binary octahedral group of quaternions preserving the quaternionic root system of $F4$⁠. The Weyl groups of many Lie algebras, such as, $G2,SO(7),SO(8),SO(9),SU(3)XSU(3)$⁠, and $SP(3)×SU(2)$ have been constructed as the subgroups of $Aut(F4)$⁠. We have also classified the other non-parabolic subgroups of $Aut(F4)$ which are not Weyl groups. Two subgroups of orders 192 with different conjugacy classes occur as maximal subgroups in the finite subgroups of the Lie group $G2$ of orders 12096 and 1344 and proves to be useful in their constructions. The triality of $SO(8)$ manifesting itself as the cyclic symmetry of the quaternionic imaginary units $e1,e2,e3$ is used to show that $SO(7)$ and $SO(9)$ can be embedded, triply symmetric way in $SO(8)$ and $F4$ in respectively.