Geodesic
The gauge-natural bilinear brackets on couples of linear vector fields and linear p-forms
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 75 (2021) no. 2, pp. 74-92.

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We give complete description of all gauge-natural bilinear operators A transforming pairs of couples of linear vector fields and linear p-forms on a vector bundle E into couples of linear vector fields and linear p-forms on E and satisfying the Jacobi identity in Leibniz form.
Keywords: Natural operator, linear vector field, linear p-form, Jacobi identity in Leibniz form 
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Kurek, Jan; Mikulski, Włodzimierz M. The gauge-natural bilinear brackets on couples of linear vector fields and linear p-forms. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 75 (2021) no. 2, pp. 74-92. http://geodesic.mathdoc.fr/item/AUM_2021_75_2_a4/

[1] Kolar, I, Michor, P. W., Slovak, J., Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993.

[2] Mikulski, W. M., The gauge-natural bilinear operators similar to the Dorfman–Courant bracket, Mediterr. J. Math. 17 (2) (2020), Art. 40, 25 pp.