This cookbook was insightful in the sense that I once tried proving several of its identities via Kronecker products but then found that it was much easier using tensor notation, so easy that I almost forgot Kronecker products altogether. I put together some proofs in my blog [0] (note: it's incomplete and the URL might change!).

[0] https://mbustamanter.github.io/ssg-blog/2022/matder1/