Real subalgebras in the matrix Lie algebra $M(2,\mathbf C)$

In this paper we classify all real subalgebras (up to the conjugation) of dimensions 5, 6, and 7 in the Lie algebra of all complex matrices of the second order. In combination with recent results by F. A. Belykh, A. Yu. Borzakov, and A. V. Loboda (Russian Mathematics (Iz. VUZ) 51 (5), 11–23 (2007)) this gives a complete classification of all subalgebras in the specified matrix Lie algebra. The description is presented in two different forms, namely, in the framework of the theory of Lie algebras and their subalgebras, on one hand, and in the matrix form, on the other hand.