Exponential map of Lie brakets

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Let $G$ be a compact matrix Lie group and $(\mathfrak{g},[,])$ its Lie algebra. For $x$, $x'$ $\in$ $\mathfrak{g}$, $[x,x']=xx'-x'x$. My question is if there exists a relation between exp$(t[x,x'])$ and exp$(tx)$, exp$(tx')$.