Chow Künneth failure for an elliptic self-product

On $E\times E$ for a complex elliptic curve, the two factor curves and the diagonal have intersection matrix $\left(\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right)$, of determinant two. Their cohomology classes are therefore independent. The degree-one image of the external product of Chow classes has cohomology contained in the span of the two factor classes. The cycle class map thus shows that the diagonal is outside that image, proving failure of surjectivity.