Geodesic
Homogeneous orthogonal decompositions of commutative algebras and Hadamard matrices
Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 4, pp. 1107-1110.

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It is shown that a commutative algebra is a free module over every its subalgebra of the family forming its homogeneous orthogonal decomposition. As a corollary the equivalence of notions of a Hadamard matrix and an orthogonal decomposition of a commutative algebra into the sum of two-dimensional subalgebras is deduced. 
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D. N. Ivanov. Homogeneous orthogonal decompositions of commutative algebras and Hadamard matrices. Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 4, pp. 1107-1110. http://geodesic.mathdoc.fr/item/FPM_1995_1_4_a20/

[1] Ivanov D. N., “Ortogonalnye razlozheniya poluprostykh assotsiativnykh algebr”, Vestnik MGU. Ser. 1. Mat., mekh., 1988, no. 1, 9–14 | MR

[2] Ivanov D. N., “Teorema o podalgebrakh, obrazuyuschikh ortogonalnoe razlozhenie assotsiativnoi algebry”, Uspekhi matem. nauk, 44:2 (1989), 231–232 | MR

[3] Kholl M., Kombinatorika, Mir, M., 1970 | MR