Realizations of subgroups of type $D_{8}$ of connected exceptional simple Lie groups of type $E_{8}$

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In [4],we realized subgroups of type A%, A4 x A4 and A2 x E$ of the compact simple Lie group of type E%. In thispaper, we shallrealize subgroups of type D% of the compact and non-compact simple Lie groups of type E%. In [5], [6], [10] and [11], Yokota and some members of his school found all involutive automorphisms a and realized subgroups Ga of fixed points of connected exceptional simple Lie groups G explicitly, which correspond to Berger's result of simple Lie algebras [2]. But in their results concerning subgroups of type Dg of Lie groups of type E%, the definition of subgroups are not clear and proof of isomorphism is very difficultin comparison with their other results. We improve those resultsin thispaper. Our improvement make results that are more simple and intelligible.Hence they are of widely applicable to symmetric spaces. Our resultsare as follows.