Let $\mathfrak{u}(n)$ be the Lie algebra of the Lie group $U(n)$. I can define a positive-definite inner product over $\mathfrak{u}(n)$ in this way: if $A,B \in \mathfrak{u}(n)$ I define $\langle A,B \rangle := \Re(\operatorname{Tr}(AB^\*))$, where by $\Re$ I denote the real part and by $B^\*$ the conjugate transpose of $B$. Why does this inner product over $\mathfrak{u}(n)$ define the unique left-invariant metric on the Lie group $U(n)$?