Lie subring whose sum with any subring is a subring not implies ideal
This article gives the statement and possibly, proof, of a non-implication relation between two Lie subring properties. That is, it states that every Lie subring satisfying the first Lie subring property (i.e., Lie subring whose sum with any subring is a subring) need not satisfy the second Lie subring property (i.e., ideal of a Lie ring)
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Let be the Klein four-group with two of the non-identity elements. The four elements of are . We define the Lie bracket as follows:
This completely determines the Lie bracket:
Let be the subring of defined as . The sum of with any subring of that does not contain is the whole ring (since is maximal as a subgroup of ). Also, the sum of with any subring of contained in is . Thus, in any case, the sum of with any subring is a subring. (for more, see fact (1)).
However, is not an ideal of , because , and is not in .