On some determinantal identities formation laws

We show that Muir's law of extensible minors, Cayley's law of complementaries and Jacobi's identity for minors of the adjugate [Determinantal identities Linear Algebra and its Applications 52/53 (1983) pp. 769-791] are equivalent. We also show our generalization of Muhlbach/Muir's extension principle [A generalization of Muhlbach's extension principle for determinantal identities. Linear and Multilinear Algebra 61 (10) (2013) pp. 1363-1376] is equivalent to its previous form derived by Muhlbach. As a corollary, we show that Muhlbach-Gasca-(Lopez-Carmona)-Ramirez identity [A generalization of Sylvester's identity on determinants and some applications. Linear Algebra and its Applications 66 (1985) pp. 221-234/On extending determinantal identities. Linear Algebra and its Applications 132 (1990) pp. 145-162] is equivalent to its generalization found by Beckermann and Muhlbach [A general determinantal identity of Sylvester type and some applications. Linear Algebra and its Applications 197,198 (1994) pp. 93-112].