We revisit the protocol for proving Elliptic Curve Inner Products (ECIPs) by Eagen. The goal of this document is to supersede and summarize prior material and discussions by Bassa and Goodell et al. In this document, we provide a self-contained explanation of the techniques, from the required algebraic geometry, to the interactive proof and its soundness, to the composition with a simulation extractable non-interactive proof and the efficiency of expressing the verifier as an R1CS circuit, resulting in the concrete gadget of Parker. Along the way, we clarify proof details, including simplifying the soundness argument for the interactive protocol, and correct other minor issues in the prior works. We aim for this document to be readable (and verifiable) by an audience with preexisting knowledge of elliptic curves, but not wider theoretical results in algebraic geometry or Galois theory. Additionally, we assume an understanding of the concepts of interactive proofs, but provide the required formal definitions along the way.