In this PhD thesis, we consider two problems that are related to finite simple groups of Lie type. First of them is a problem mentioned in the Kourovka notebook: describe the finite simple groups in which every element is a product of two involutions. We consider the simple orthogonal groups in even characteristic, and solve the problem for them. Since other groups have been dealt with elsewhere, the problem is then solved completely. The second part of the thesis is related to Lie algebras. Every complex simple Lie algebra has a compact real form that is associated with a compact Lie group. In this thesis, we consider the Lie algebra of type E8, and give a new construction of its compact real form. The Lie product is defined using the irreducible subgroup of shape 25+10 ·GL5(2) of the automorphism group.