semi-simple Lie groups endowed with a left invariant metric isometric to a 7-sphere and Lie group with a bi-invariant metric

Question

Let $(G,g)$ be a semi-simple compact Lie group endowed with a left invariant metric, $(H,h)$ a compact semi-simple Lie group endowed with a bi-invariant and $(S^7,can)$ the 7-sphere endowed with its canonical metric. I came across a situation where there is an isometry $\phi:(G,g)\rightarrow (H,h)\times (S^7,can)$. I have many reasons to thank that such a situation is not possible but I can't prove it. May be someone can help me. Thank you.