motives - Kunneth formula for motivic cohomology

I now remember a nice argument, why there's no Kunneth formula for Chow groups of $X times X$ unless $X$ has a Tate motive. Let $X$ be smooth projective of dimension $d$.
We start with a decomposition of a diagonal:

$$[Delta] = sum_{i,j} alpha^i_j beta^{d-i}_j in oplus_i CH^i(X) otimes CH^{d-i}(X)$$
We can assume $alpha^i_j$ are linearly independent.
In this case we can show that $alpha^i_j$ form a basis of Chow groups and
$beta^{d-i}_j$ is the dual basis.

Indeed, as a correspondence $[Delta]$ acts as identity on Chow groups,
so for any class c,

$$c = [Delta]c = sum_{i,j} alpha^i_j deg(beta^{d-i}_j cup c),$$
and the claim follows if we substitute $c = alpha^i_j$.

Now $CH_i(X) = Hom(mathbb Z(i)[2i], M(X))$ and we can consider the set of $alpha^i_j$ as a morphism of motives

$$oplus_{i,j}mathbb Z(i)[2i] to M(X).$$
A simple computation shows that it is an isomorphism with the inverse given by $beta^i_j$.

And of course, on the other hand, if $X$ has a Tate motive, then Kunneth formula for Chow groups follows (it doesn't answer the question, since I only consider smooth projective varieties).

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