an example of automorphisms of Lie algebras: not clear the upshot I was following Humphreys Lie algebra, and in first chapter, I came across following example: > $, which is $\sigma$; it is an element of $SL(3,F)$. And other is the element $s$ in $SL(2,F)$ shown as matrix in last part. I didn't get what exactly author want to state here? Can one clarify a little the upshot of this para?

The question what the author wants to say with this example has no unique answer. I think he wants to introduce the automorphism group ${\rm Aut}(L)$ for (simple) Lie algebras $L$ and the ideal ${\rm Inn}(L)$ of inner automorphisms, based on the example $L=\mathfrak{sl}_2(F)$. One "upshot" is the relation $(*)$, but in general it is a standard result in a course on Lie algebras to determine the groups ${\rm Aut}(L)$ for all simple Lie algebras. For example, we have the following result (see Jacobson's book on Lie algebras, section $IX$, Theorem $5$, or this MSE-question):

_Theorem 5:_ The group of automorphisms of the simple Lie algebra $L=\mathfrak{sl}_n(F)$ for a field $F$ of characteristic zero is, for $n>2$, the set of mappings $X\mapsto A^{-1}XA$ and $X\mapsto A^{-1}X^TA$. For $n=2$ all automrophisms are of the form $X\mapsto A^{-1}XA$, i.e., they are all inner.