The authors completely determine the fine gradings on the exceptional simple Lie superalgebras over an algebraically closed field of characteristic \(0\), in Kac’s classification in 1977. As it is well known, a grading (in fact, an abelian group grading) on a Lie algebra is said to be fine if it admits no proper refinements (it implies that any grading is obtained as a coarsening of some fine gradings) and the type of a grading on finite-dimensional Lie algebra \(\mathcal{A}\) is the sequence of numbers \((n_1, n_2, \dots, n_r)\) where \(n_r\) is the number of homogeneous components of dimension \(i\), \( i = 1, \dots, r\), with \(n_r \neq 0\). Thus, \(\dim \mathcal{A} = \sum_{i = 1}^r i \, n_i\). The definitions on gradings on algebras carry over in a straightforward way to superalgebras.
In this way, the authors deal in the paper with the gradings in simple Lie superalgebras \(F(4)\) and \(G(3)\) and with those with corresponding maximal abelian diagonalizable subgroups fixing the three simple ideals of \(D(2,1,\alpha)\), \(\alpha \neq 0, -1\). They obtain that, up to equivalence, there are five gradings on \(F(4)\), only two on \(G(3)\) and five, six and four, respectively, on \(D(2,1,\alpha)\), depending on the values of \(\alpha\).