Group cohomology homomorphism action under natural embedding $O(n) \times O(k) \to O(n+k)$

Question

Let we have embedding $O(n) \times O(k) \to O(n+k)$ that induces homomorphism on cohomology $H^\cdot(BO(n+k),\mathbb{Z}_2) \to H^\cdot(BO(n),\mathbb{Z}_2) \otimes H^\cdot(BO(k),\mathbb{Z}_2)$ or, in another words $\mathbb{Z}_2[w_1,...,w_{n+k}] \to \mathbb{Z}_2[p_1,...,p_n] \otimes \mathbb{Z}_2[z_{1},...,z_{k}] = \mathbb{Z}_2[p_1,...,p_n,z_1,...,z_{k}]$ ($\deg p_i = i, \deg z_i = i, \deg w_i = i$). But how that homomorphism acts?