My question is if there is a way to construct a factorization of SU(3) into products of the 3 subgroups copies of SU(2) through the root system? 
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user30656
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5No, $SU(2)$ is a simple Lie group. – Angina Seng Jan 17 '18 at 18:32
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ok thank you, could you explain a bit more? – user30656 Jan 17 '18 at 23:22
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1Rather, $SU(3)$ is a simple Lie group. A simple Lie group cannot be represented as a direct product of two Lie groups of positive dimension. – Moishe Kohan Jan 18 '18 at 00:01