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Define a left-normed commutator $[a_1, a_2, \\dots , a_n]$ inductively by $[a_1, a_2] = a_1 a_2 - a_2 a_1$, $[a_1, \\dots , a_{n-1}, a_n] = [[a_1, \\dots , a_{n-1}], a_n]$ $(n \\ge 3)$. For $n \\ge 2$, let $T^{(n)}$ be the two-sided ideal in $K \\langle X \\rangle$ generated by all commutators $[a_1,a_2, \\dots , a_n]$ $( a_i \\in K \\langle X \\rangle )$.\n  It can be easily seen that the ideal $T^{(2)}$ is generated (as a two-sided ideal"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1306.4294","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-06-18T18:44:38Z","cross_cats_sorted":[],"title_canon_sha256":"5d4d23794678c331e021f2f24e5dde8f5fd7a2c7844e6c85eba3d014614330af","abstract_canon_sha256":"c410ac753e90aa558a861ce98e97a85f8420b06023f40ab76c706806549157bf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:13:48.764009Z","signature_b64":"Sv1XMJwFTJKazN+JNkO28BhdmIAWCwZf7I/0D91V8zc6aSpGiHIsE2K7kLSHqXXauvDRL3N16tF6aIp2ZjEgBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9a020841ed7f96c607d914f04defd2e7ca6cd422e7fce79b7c588d39238c1513","last_reissued_at":"2026-05-18T00:13:48.763363Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:13:48.763363Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Relations in universal Lie nilpotent associative algebras of class 4","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Alexei Krasilnikov, Eudes Antonio da Costa","submitted_at":"2013-06-18T18:44:38Z","abstract_excerpt":"Let $K$ be a unital associative and commutative ring and let $K \\langle X \\rangle$ be the free unital associative $K$-algebra on a non-empty set $X$ of free generators. 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