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A = A1A2A1 ′, A1, A1 ′ ∈ SO1(2), A2 ∈ SO2(2) where SOk(2) = {A ∈ SO(3) |Aek = ek}(k = 1, 2) ([1]). In the present paper, we shall show firstly that the similar results hold for the groups SU(3), and Sp(3) (Theorem 1). Secondly, we shall show that any element α of the simply connected compact Lie group F4 (resp. E6 ) can be represented as α = α1α2α1 ′, α1, α1 ′ ∈ Spin1(9), α2 ∈ Spin2(9) (resp.α1, α1 ′ ∈ Spin1(10), α2 ∈ Spin2(10)) where Spink(9) = {α ∈ F4 |αEk = Ek} (resp. Spink(10) = {α ∈ E6 |αEk = Ek} (Theorem 5 (resp. Theorem 7))). Lastly, we shall show that any element α of the simply connected compact Lie group E7 can be represented as α = α1α2α1 ′α2 ′α1 ′′, α1, α1 ′, α1 ′′ ∈ Spin1(12), α2, α2 ′ ∈ Spin2(12) where Spink(12) = {α ∈ E7 |ακk = κkα, αμk = μkα} (Theorem 10). In this paper we follow the notation of [2].

Spinor-generators of compact exceptional Lie groups $F_{4}$, $E_{6}$ and $E_{7}$