Division algebras with integral elements

Pairing two elements of a given division algebra furnished with a multiplication rule leads to an algebra of higher dimension restricted by 8. This fact is used to obtain the roots of SO(4)and SP(2) from the roots ⫦ of SU(2) and the weights⫦ ½ its spinor representation. The root lattic of SO(8) described by 24 integral quaternions are obtained by pairing two sets of roots of SP(2). The root system of F₄ is constructed in terms of 24 integral and 24‘half integral’ quaternions. Th root lattice of E, expressed as 240 integral octonions are obtainedby pairing two sets of roots o F₄. Twenty four integral quaternions of SO(8) forming a discrete subgroup of SU(2) are shown to be the automorphism group of the root latticesof SO(8), F₄ and E₈. The roots of maximal subgroups SO 16), E₈ Ö SU(2), E₈ Ö SU(3), SU(9) and SU(5) Ö SU(5) of E₈ are identified witha simple method. Subsets of the discrete subgroup of SU(2) leaving maximal subgroups of E, are obtained. Constructions of E, root lattice with integral octonions in seven distinct ways are made. Magic squares of integral lattices of Goddard, Nahm, Olive, Ruegg and Schwimmer are derived. Possible physical applications are suggested.