When $n>3$ is even, how can I show that $PGL(n,\mathbb{R})$ has a faithful adjoint representation? Of course when n is even, $PGL(n,\mathbb{R})$ is not connected.
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Suppose $A \in PGL_n(\mathbb{R})$ lies in the kernel of the adjoint representation. Then for any lift $\tilde{A}$ of $A$ in $GL_n(\mathbb{R})$, and any traceless matrix $B$, we have $\tilde{A}B = B\tilde{A}$. This implies $\tilde{A}$ is scalar, and hence $A$ is the identity. 

