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Let $C$ be an associative commutative ring with 1. If $a\in C$, then $aC$ denotes the principal ideal generated by $a$. Let $l,m,n$ be nonzero elements of $C$ such that $mn\in lC$. The set of matrices $\left(\begin{array}{ccc}{a}_{11}& a_{12}{a}_{21}& -a_{11}\end{array}\right)$, where $a_{11}\in lC$, $a_{12}\in mC$, $a_{21}\in nC$, forms a Lie ring under Lie multiplication and matrix addition. The paper studies properties of these Lie rings.

Let $R$ be an associative ring with 1 and $R^{\times }$ the multiplicative group of invertible elements of $R$. In the paper, subgroups of $R^{\times }$ which may be regarded as analogues of the similitude group of a non-degenerate sesquilinear reflexive form and of the isometry group of such a form are defined in an abstract way. The main result states that a unipotent abstractly defined similitude must belong to the corresponding abstractly defined isometry group.

Let $K$ be an associative and commutative ring with $1$, $k$ a subring of $K$ such that $1\in k$, $n\ge 2$ an integer. The paper describes subrings of the general linear Lie ring $g{l}_n\left(K\right)$ that contain the Lie ring of all traceless matrices over $k$.