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We develop a new approach to find asymptotic symmetries in general relativityas a modification of the Lie-algebra-based approach proposed in T. Tomitsuka etal. [Classical Quantum Gravity 38, 225007 (2021)]. Those authors proposed analgorithmic protocol to investigate asymptotic symmetries. In particular, theirguiding principle helps us to find a non-vanishing charge that generates aninfinitesimal diffeomorphism. However, in order to check the integrabilitycondition for the charges, it is necessary to solve differential equations toidentify the integral curve of vector fields, which is usually quite hard. Inthis paper, we provide a sufficient condition of the integrability conditionthat can be checked without solving any differential equations, avoiding thedifficulties in the approach in the above reference. As a demonstration, weinvestigate the asymptotic symmetries on a Killing horizon and find a new classof asymptotic symmetries. In 4D spacetimes with a spherical Killing horizon, weshow that the algebra of the corresponding charges is a central extension ofthe algebra of vector fields.

Modification of a Lie-algebra-based approach and its application to asymptotic symmetries on a Killing horizon