An excellent reference for this kind of descriptions (and for many other facts about $E_8$!) is Skip Garibaldi's paper "$E_8$, the most exceptional algebraic group" in the Bulletin of the AMS (here). In particular, in section 4, he describes various possible branchings, and Example 4.4 deals with the branching $A_8 \subset E_8$.

The specific example is rather short, but he includes some further references that might be helpful.

An excellent reference for this kind of descriptions (and for many other facts about $E_8$!) is Skip Garibaldi's paper "$E_8$, the most exceptional algebraic group" in the Bulletin of the AMS (here). In particular, in section 4, he describes various possible branchings, and Example 4.4 deals with the branching $A_8 \subset E_8$.

The specific example is rather short, but he includes some further references that might be helpful:

• Hans Freudenthal, "Sur le groupe exceptionnel $E_8$", Nederl. Akad. Wetensch. Proc. Ser. A 56 (=Indag. Math. 15) (1953), 95–98 (=pages 284–287 in his Selecta published by the EMS in 2009).

• William Fulton & Joe Harris, Representation Theory: A First Course (Springer 1991, GTM 129), exercise 22.21 on page 361.

• John Faulkner, "Some forms of exceptional Lie algebras", Comm. Algebra 42 (2014), 4854–4873 (=arXiv:1305.0746).

[Freudenthal's paper is completely explicit and elementary; Fulton & Harris provide a bit more theoretical background; Faulkner is more general and works over an arbitrary commutative ring. —Gro-Tsen]