Show that for $(A,*)$ algebraic system $a*(a*b) = a*b$

Question

Suppose we are given that for algebraic system $(A,*)$ that $$(a*b)*a = a $$ and $$(a*b)*b = (b*a)*a$$ then how can I show that $$a*(a*b) = a*b$$ such that $a,b$ belongs to $A$?

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I know that $(a*b)$ is left identity from first relation but what conclusion I can draw from second given relation to show required relation? I tried to reach to any of two given relation from required relation by

$$a*(a*b)*a = (a*b)*a$$ $$a*a = a$$

Now what? this could have been proved if $a$ were identity but that does not seem to be case here.

Answer 1

There is no way you can prove it, because it is false. Take the semigroup with zero $S = \{a, b, ab, ba, 0\}$ defined by the relations $aba = a$, $bab = b$, $aa = 0$ and $bb = 0$. Then $(ab)a = a$, $(ab)b = (ba)a = 0$, but $a(ab) = 0 \not= ab$.

Answer 2

Simple!

a*(ab)=[(ab)a](ab)=ab

Notice that we only need to use the first identity twice.The rest of other two are useless!