Geodesic
Semisimple automorphism groups of four-dimensional affine space
Izvestiya. Mathematics , Tome 23 (1984) no. 1, pp. 171-183

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It is shown that any regular action of a semisimple algebraic group on $\mathbf A^4$ is equivalent to a linear action. Bibliography: 22 titles. 
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D. I. Panyushev. Semisimple automorphism groups of four-dimensional affine space. Izvestiya. Mathematics , Tome 23 (1984) no. 1, pp. 171-183. http://geodesic.mathdoc.fr/item/IM2_1984_23_1_a4/