Higher dualisations of linearised gravity and the $$ {A}_1^{+++} $$ algebra

The non-linear realisation based on $A_1^{+++}$ is known to describe gravity in terms of both the graviton and dual graviton. We extend this analysis at linearised level find equations motion for first higher description that it contains. also give a systematic method finding additional fields beyond those are required construct actions all possible descriptions realisation. show these closely correlated with second fundamental representation $A_1^{+++}\,$.