New solutions of the Yang–Baxter equation with a square

The paper is devoted to the Yang–Baxter equation with the square, that is, to the equation
$$ R([R(a),b]-[R(b),a])=R^2([a,b])+[R(a),R(b)], $$
where $a,b\in g$, $g$ – is a Lie algebra, and $R$ is a linear operator on the vector space $g$. Two series of operators $R$, satisfying this equation are constructed. In the construction we use Lie subalgebras in the matrix algebra, complementary to the subspace of matrices with zero last row.