Jos´ e de Jes´ us Cruz Guzm´ an and Zbigniew Oziewicz FOUR MAXWELL’S EQUATIONS FOR NON-INERTIAL OBSERVER AND FR ¨ OLICHER-NIJENHUIS ALGEBRA Abstract. Fr¨ olicher-Nijenhuis and Schoten-Nijenhuis graded Lie modules, are applied for derivation explicitly observer-dependent four Maxwell equations. Observer field is almost product structure (1, 1)-tensor field, that is idempotent operator giving a (1+3)-split. In present pa- per an observer field is not restricted to be neither inertial nor holonomic, and need not to be orthogonal. Contents 1. No Measurement without Observer 108 1.1. Axiomatic electromagnetics 108 1.2. No measurement without an observer field 110 1.3. Notation 111 1.4. Technical tool 112 1.5. Relation to earlier publications 115 1.6. Advantages of axiomatic approach 116 2. Graßmann algebra 117 3. Differential Graded Algebra Graded Lie Algebra 120 4. Universal Graßmann module 121 4.1. Nonassociative F -algebra structure on Graßmann module 122 5. Leibniz/Loday and Gerstenhaber algebra 122 6. Fr¨olicher and Nijenhuis decomposition [1956] 124 6.1. Universal property of d ∈ der(F ,M ) 124 6.2. Fr¨olicher and Nijenhuis decomposition [1956] 125 7. Main definition 126 8. Theorem and proof 128 Date : March 3, 2003. Bulletin de la Soci´ et´ e de Sciences et des Lettres de  L´ od´ z, Volume LIII, S´ erie: Recherches sur les D´ eformations, Volume XXXIX (2003) pp. 107–140; PL ISSN 0459-6854. Key words and phrases. 1+3-split, accelerated observer, anholonomic observer, non-orthogonal observer, four Maxwell equations. 107