On page 627 of Vector Fields on Spheres, Adams writes:
Our first concern is to show that the following exact sequence splits. $$\mathbb Z = \tilde K_{\mathbb R}(\mathbb R P^{4t}/\mathbb RP^{4t-1}) \xleftarrow{i} \tilde K_\mathbb R (\mathbb RP^n/\mathbb RP^{4t-1}) \xleftarrow{j} \tilde K_\mathbb R(\mathbb RP^n/\mathbb RP^{4t}).$$
I understand the definition of a split short exact sequence. However, I cannot find anything about a split sequence of length three. What does Adams mean by 'the following exact sequence splits'?
It appears that to prove this claim, Adams shows $j$ is a monomorphism and $i$ is an epimorphism. This implies that $$0 \leftarrow \mathbb Z = \tilde K_{\mathbb R}(\mathbb R P^{4t}/\mathbb RP^{4t-1}) \xleftarrow{i} \tilde K_\mathbb R (\mathbb RP^n/\mathbb RP^{4t-1}) \xleftarrow{j} \tilde K_\mathbb R(\mathbb RP^n/\mathbb RP^{4t}) \leftarrow 0$$ is exact.
A sequence $A \to B \to C$ is a split exact sequence if it can be extended to a short exact sequence $0 \to A \to B \to C \to 0$ that splits.
As stated in the question, Adams shows that we can extend to an exact sequence $$0 \leftarrow \mathbb Z = \tilde K_{\mathbb R}(\mathbb R P^{4t}/\mathbb RP^{4t-1}) \xleftarrow{i} \tilde K_\mathbb R (\mathbb RP^n/\mathbb RP^{4t-1}) \xleftarrow{j} \tilde K_\mathbb R(\mathbb RP^n/\mathbb RP^{4t}) \leftarrow 0.$$
This sequence splits follows from the following easy lemma: any SES $0 \to A \xrightarrow{j} B \xrightarrow{i} \mathbb Z \to 0$ splits. Since $i$ is surjective, chose some $b \in B$ that maps to 1. Then define a section $s : \mathbb Z \to B$ by $s(1) = b$.