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This includes, as a particular case, $\\exp(X) \\exp(Z)$, with $[X,Z]$ containing other elements in addition to $X$ and $Z$. The algorithm exploits the associativity of the BCH formula and is based on the decomposition $\\exp(X)\\exp(Y)\\exp(Z)=\\exp(X)\\exp({\\alpha Y}) \\exp({(1-\\alpha) Y}) \\exp(Z)$, with $\\alpha$ fixed in such a way that it reduces to $\\exp"},"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":"1503.08198","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-03-27T19:37:47Z","cross_cats_sorted":["hep-th","math.MP","math.RT","quant-ph"],"title_canon_sha256":"f39c9df1b22af3aeca79795de385b5dd464e821dc318e82a135721fd5a50194b","abstract_canon_sha256":"943972544e5a54ec5db6042a4a8d5d4c942ccaa487fdeb70d53fab85256d0135"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:25.624763Z","signature_b64":"9ekQllw2SDTa0Wd5jUgrcpvtpzfhega8Zww6Ige+v78cJ2UOfojP5HwE8P7J3CO/KA7RNa9sPLwkksYKJVC9Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"63dcf081d919465d19c23addb0fea939824930c21ddc048e77f307c4256f7d5d","last_reissued_at":"2026-05-18T01:36:25.624231Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:25.624231Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Classification of Commutator Algebras Leading to the New Type of Closed Baker-Campbell-Hausdorff Formulas","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math.MP","math.RT","quant-ph"],"primary_cat":"math-ph","authors_text":"Marco Matone","submitted_at":"2015-03-27T19:37:47Z","abstract_excerpt":"We show that there are {\\it 13 types} of commutator algebras leading to the new closed forms of the Baker-Campbell-Hausdorff (BCH) formula $$\\exp(X)\\exp(Y)\\exp(Z)=\\exp({AX+BZ+CY+DI}) \\ , $$ derived in arXiv:1502.06589, JHEP {\\bf 1505} (2015) 113. This includes, as a particular case, $\\exp(X) \\exp(Z)$, with $[X,Z]$ containing other elements in addition to $X$ and $Z$. The algorithm exploits the associativity of the BCH formula and is based on the decomposition $\\exp(X)\\exp(Y)\\exp(Z)=\\exp(X)\\exp({\\alpha Y}) \\exp({(1-\\alpha) Y}) \\exp(Z)$, with $\\alpha$ fixed in such a way that it reduces to $\\exp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.08198","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1503.08198","created_at":"2026-05-18T01:36:25.624313+00:00"},{"alias_kind":"arxiv_version","alias_value":"1503.08198v3","created_at":"2026-05-18T01:36:25.624313+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.08198","created_at":"2026-05-18T01:36:25.624313+00:00"},{"alias_kind":"pith_short_12","alias_value":"MPOPBAOZDFDF","created_at":"2026-05-18T12:29:32.376354+00:00"},{"alias_kind":"pith_short_16","alias_value":"MPOPBAOZDFDF2GOC","created_at":"2026-05-18T12:29:32.376354+00:00"},{"alias_kind":"pith_short_8","alias_value":"MPOPBAOZ","created_at":"2026-05-18T12:29:32.376354+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MPOPBAOZDFDF2GOCHLO3B7VJHG","json":"https://pith.science/pith/MPOPBAOZDFDF2GOCHLO3B7VJHG.json","graph_json":"https://pith.science/api/pith-number/MPOPBAOZDFDF2GOCHLO3B7VJHG/graph.json","events_json":"https://pith.science/api/pith-number/MPOPBAOZDFDF2GOCHLO3B7VJHG/events.json","paper":"https://pith.science/paper/MPOPBAOZ"},"agent_actions":{"view_html":"https://pith.science/pith/MPOPBAOZDFDF2GOCHLO3B7VJHG","download_json":"https://pith.science/pith/MPOPBAOZDFDF2GOCHLO3B7VJHG.json","view_paper":"https://pith.science/paper/MPOPBAOZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1503.08198&json=true","fetch_graph":"https://pith.science/api/pith-number/MPOPBAOZDFDF2GOCHLO3B7VJHG/graph.json","fetch_events":"https://pith.science/api/pith-number/MPOPBAOZDFDF2GOCHLO3B7VJHG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MPOPBAOZDFDF2GOCHLO3B7VJHG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MPOPBAOZDFDF2GOCHLO3B7VJHG/action/storage_attestation","attest_author":"https://pith.science/pith/MPOPBAOZDFDF2GOCHLO3B7VJHG/action/author_attestation","sign_citation":"https://pith.science/pith/MPOPBAOZDFDF2GOCHLO3B7VJHG/action/citation_signature","submit_replication":"https://pith.science/pith/MPOPBAOZDFDF2GOCHLO3B7VJHG/action/replication_record"}},"created_at":"2026-05-18T01:36:25.624313+00:00","updated_at":"2026-05-18T01:36:25.624313+00:00"}