Lemma 45.4.6. The category $M_ k$ is additive.

**Proof.**
Let $(Y, p, m)$ and $(Z, q, n)$ be motives. If $n = m$, then a direct sum is given by $(Y \amalg Z, p + q, m)$, with obvious notation. Details omitted.

Suppose that $n < m$. Let $X$, $c_2$ be as in Example 45.3.7. Then we consider

where we have used Lemma 45.4.4. This reduces us to the case discussed in the first paragraph. $\square$

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