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While it is well known that the octonions can be described as a twisted group algebra, our purpose is to point out that this is a useful description. We show how the basic properties of the octonions follow easily from our definition. We give a uniform description of the sixteen orders of integral octonions containing the Gravesian integers, and a co"},"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":"1702.05705","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-02-19T06:27:23Z","cross_cats_sorted":[],"title_canon_sha256":"77871a89b35712b4119a7aa614eeeec07b9841a771db8fac2476269aa850b0ac","abstract_canon_sha256":"399b00efb85bd07a52b2a9e1386dbfcff48dedf8fa8c1d07aad714a503f0f036"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:50:25.585661Z","signature_b64":"fx383Q4XkGLQK5a/kzgOAGLgfGKJADc7tx6bxpmY/PHDFnOE07hIEBdN/IWE+VJqm2NzVzVRcW1ZZKqqC12aBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"80d0f4185abc28ad921e19918163e51aa2effc7520edad7c76deb25d2982b217","last_reissued_at":"2026-05-18T00:50:25.584886Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:50:25.584886Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The octonions as a twisted group algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Tathagata Basak","submitted_at":"2017-02-19T06:27:23Z","abstract_excerpt":"We show that the octonions can be defined as the $\\mathbb{R}$-algebra with basis $\\lbrace e^x \\colon x \\in \\mathbb{F}_8 \\rbrace$ and multiplication given by $e^x e^y = (-1)^{\\varphi(x,y)}e^{x + y}$, where $\\varphi(x,y) = \\operatorname{tr}(y x^6)$. While it is well known that the octonions can be described as a twisted group algebra, our purpose is to point out that this is a useful description. We show how the basic properties of the octonions follow easily from our definition. 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