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In this short paper, we look into a conclusion drawn by Alzalg (J Optim Theory Appl 169:32-49, 2016). We think the conclusion drawn in the paper is incorrect by pointing out three things. First, we provide a counterexample that the proposed inner product does not satisfy bilinearity. Secondly, we offer an argument why a pth-order cone cannot be self-dual under any reasonable inner product structure on R n . Thirdly, even under the assumption that all elements operator commute, the inner product becomes an official inner product and the arbitrary-order cone can be shown as a symmetric cone, we think this condition is still unreasonable and very stringent so that the result can only be applied to very few cases. Keywords pth-order cone • Second-order cone • Inner product • Jordan algebras Mathematics Subject Classification 17C10 • 52A07 Communicated by Sándor Zoltán Németh.B Yen-chi Roger Lin

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In this paper we show that the conjecture of Lemmens and Seidel of 1973 for systems of equiangular lines with common angle arccos(1/5) is true. Our main tool is forbidden subgraphs for smallest Seidel eigenvalue −5.