group theory - Show that the fundamental representation is a representation

I want to see that the fundamental representation is a representation. Suppose the structure constants $f^{abc}$ are given. We can assume there is at least one non-zero structure constant, otherwise any set of commuting $d \times d$ Hermitian matrices would comprise a representation. So assume that $a,b,c$ are such that $f^{abc} \neq 0$. According to Peskin & Schroeder (1995), the $k$-dimensional vector $\xi^n$ is the (1-dimensional) fundamental representation over some field $\mathbb{F}$. If so then $$ i f^{abc} \xi^c \stackrel{!}{=} [\xi^a,\xi^b] = \xi^a\xi^b - \xi^b\xi^a = \xi^a\xi^b - \xi^a\xi^b = 0 $$ since $\xi^n \in \mathbb{F}$ for each $n$, thus $\xi^a$ commutes with $\xi^b$. This is a contradiction since we assumed that there exists at least one $f^{abc} \neq 0$.

(1) What is wrong?

(2) How can we express $\xi^a$ in terms of $f^{abc}$, such that we see that the fundamental representation is indeed a representation?

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EDIT: The pages in Peskin are 498-499. According to Peskin page 498 the definition of a d-dimensional representation is a set of $d \times d$ Hermitian matrices $\xi^a$ such that for given structure constants $f^{abc}$, $if^{abc}\xi^c = [\xi^a,\xi^b]$. According to Peskin page 499, in the fundamental representation of a Lie group of dimension $k$, there exists some field $\mathbb{F}$ such that the $k$-dimensional vector over $\mathbb{F}$ is the fundamental representation.