The canonical line bundle on $\mathbb{C}P^{n}$ is denoted by $\ell_{n}$. It is well known that: $$(\ell_{1}\otimes \ell_{1})\oplus1 \simeq\ell_{1}\oplus\ell_{1}$$ Both sides are isomorphic bundles as two topological vector bundles over $\mathbb{C}P^{1}$.

(A topological question) How can we compare $(\ell_{n}\otimes \ell_{n})\oplus1 $ with $\ell_{n}\oplus\ell_{n}$. We know that their restriction to $\mathbb{C}P^{1}\subset \mathbb{C}P^{n}$ are isomorphic. So what is the difference element as an element of the relative K-theory $K(\mathbb{C}P^{n},\mathbb{C}P^{1})$?

(An algebraic geometric question) Is the identity $$(\ell_{1}\otimes \ell_{1})\oplus1 \simeq\ell_{1}\oplus\ell_{1}$$ valid as line bundles in the context of algebraic geometry?