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For simplicity let us work in {SL_2(k)}, which is slightly simpler because all semisimple (which, in this linear context, simply means diagonalisable) elements other than {\pm 1} are regular, which in the case of linear groups just means that the eigenvalues are distinct. Every regular element {a} of the three-dimensional {SL_2(k)} then generates a one-dimensional maximal torus {Z(a)}, which is also the centraliser of {a} (the set of all matrices in {SL_2(k)} that commute with {a}).

Let {A} be a {K}-approximate group that generates {SL_2(k)}, where we think of {K} as being small, say {K=O(1)} to simplify the discussion (of course, in the full argument we will need to track the dependence on {K} and keep it polynomial in nature). We may assume that {A} is not too small (more precisely, {|A| \geq CK^C} for some large {C}). As {A} lives in the three-dimensional group {SL_2(k)}, it is reasonable to expect that the intersection of {A} with a one-dimensional subset, such as a maximal torus, would be of size about {|A|^{1/3}}. And indeed this is true, as was observed by Helfgott:

Lemma 2 (Intersection with torus) If a maximal torus {T} intersects {A^2} (say) outside of {\{-1,+1\}}, then {A^{20} \cap T} (say) has cardinality {\sim |A|^{1/3}}.

For more general Lie groups, one can establish a similar upper bound (for more general algebraic varieties than just a torus) by using the Larsen-Pink inequality, which I discussed in this previous blog post. The lower bound is more important to us; it comes from noting that the conjugacy class {\{ gag^{-1}: g \in A^{20} \}} lies in {A^{41}} and also in a two-dimensional subset {C(a)} of {SL_2(k)}, and so should have cardinality {O(|A|^{2/3})}.

This lemma gives an important gap property: as soon as a maximal torus encounters just one regular element of {A^2}, it in fact has to absorb quite a lot of elements of the slightly larger set {A^{20}}. It is this gap which we exploit as follows. Let us say that a maximal torus {T} is involved if it intersects {A^2} outside of {\{-1,+1\}}.

Lemma 3 (Pivot rotation lemma) If {T} is an involved torus, then {aTa^{-1}} is also an involved torus for any {a \in A}.

Proof: (Sketch) We conjugate {aTa^{-1}} by a further element {b \in A}, and then multiply it on the left to get a new torus {cbaTa^{-1}b^{-1}}, where {a, b \in A}. On the one hand, we can think of this torus as of the form {xTy}, where {x \in A^3/T} and {y \in T \backslash A^2}. From Lemma 2 we see that there are only {O(|A|^{2/3})} values of {x} and {y}, and so there are {O(|A|^{4/3})} tori here. On the other hand, there are {|A|^2} choices of {b} and {c}. Hence there must be lots of collisions of the form

\displaystyle  cbaTa^{-1}b^{-1} = c' b' aTa^{-1} (b')^{-1}.

Taking quotients, we see that

\displaystyle  baTa^{-1}b^{-1} = b' aTa^{-1} (b')^{-1}

and thus {b' b^{-1}} lies in the normaliser {N(aTa^{-1})} of {aTa^{-1}}. But this is only twice as large as {aTa^{-1}} (the quotient of {N(aTa^{-1})} by {aTa^{-1}} is the Weyl group of {SL_2}, which in this two-dimensional case has cardinality {2!}.) But because there are so many collisions, it is not hard to use a pigeonhole argument to find a non-trivial pair {b,b'} where {b' b^{-1}} lies in the torus {aTa^{-1}} itself, and is not equal to {\pm 1}. This makes {aTa^{-1}} an involved torus as required. \Box

Now suppose that {A} generates all of {SL_2(k)}, then the set of involved tori is then invariant under conjugation by arbitrary elements of {SL_2(k)}. But all maximal tori in {SL_2} are conjugate to each other, and it is not hard to show that any large {A} must intersect at least one maximal torus non-trivially (using something called an escape from subvarieties argument), and so every maximal torus is an involved torus. But then there are {\sim |SL_2(k)|^{2/3}} such tori; this is only consistent with Lemma 2 if {|A| \sim |SL_2(k)|}, at which point one is done. A slightly more sophisticated version of this argument also works when {A} does not generate all of {SL_2(k)}.

It is instructive to compare the above argument to the analogous sum-product argument. Let {A} be an approximate subring of a field {F} (let us not define this concept precisely here). We say that a non-zero field element {\xi} is involved with {A} if the set of sums {\{ a+\xi b: a, b \in A \}} are not distinct, i.e. {|A + \xi A| < |A|^2}. The analogue of Lemma 2 here is that if {\xi} is involved, then in fact {\xi} must lie in the quotient set {Q := (A-A) / ((A-A) \backslash \{0\})} of {A}, just by using a collision {a+\xi b = c + \xi d} to solving for {\xi}. The analogue of Lemma 3 is then the observation that if {\xi} and {\eta} lie in {Q}, then {\xi+\eta} and {\xi \cdot \eta} are still sufficiently involved with {A} that one can bound {|A + (\xi+\eta) A|} or {|A + \xi \eta A|} to be strictly smaller than {|A|^2}, so that the sum and product of any two involved elements is again involved; this can be deduced from the so-called Katz-Tao lemma, which roughly asserts that if a set {A} is approximately closed under sums and products, then it (or a large portion thereof) is closed under more complicated polynomial and rational operations as well.