I'm trying to understand the following theorem about the curvature tensor of a symmetric space:
Let $R$ the curvature tensor of the space $G/K$ corresponding to the Riemannian structure $Q$ ,then at the point $o=\pi(0) \in G/K$
$R_o(X,Y)Z=-\left[\left[X,Y\right],Z\right]$ for any $X,Y,Z \in T_0(G/K)$
My concern is that if i consider a compact semisimple Lie group there is a natural structure of symmetric group with Riemannian structure given by the Cartan-Killing metric, that can be represented as $g_{ij}=c_{im}^n c^m_{jn}$ where $c_{ij}^k$ are the structure constant of the Lie algebra associated to the Lie group.
Such metric is left invariant and the multiplication is a transitive so in a chart i have $g(X,Y)_x=g(L_x X_e, L_x Y_e)=g(X_e,Y_e)$ for any left invariant vector field. So the metric is constant and that implies that the Christoffel symbols of such metric are $0$ and i'm expecting the curvature tensor to be zero as well, but this would means that $\left[\left[\mathfrak{g},\mathfrak{g}\right],\mathfrak{g}\right]=0$ that is impossible beacause the Lie algebra is semisimple and so not solvable.
The part in bold is where i think there is an error, but i can't figure out what this error is