Removing aluminum oxide from impure silica
You may try the famous Bayer process, which is used for extracting alumina form bauxite. When treated at $160°$C and high pressure with $\ce{NaOH}$, a mixture like yours dissolves $\ce{SiO2}$ and $\ce{Al2O3}$, and lets the other compounds insoluble. This is a first purification. $$\ce{SiO2 + 2 NaOH -&gt;  Na2SiO3  +  H2O}$$ $$\ce{Al2O3  +  2 NaOH + 3 H2O -&gt; 2 Na[Al(OH)4]}$$Later on, bubbling $\ce{CO2}$ in the solution makes $\ce{Al(OH)3}$ to precipitate. $$\ce{Na[Al(OH)4] + CO2  -&gt; Al(OH)3(s) + NaHCO3}$$ Of course $\ce{NaHCO3}$ is immediately transformed into $\ce{Na2CO3}$ by $\ce{NaOH}$. $\ce{Al(OH)3}$ can be eliminated by filtration. And the final solution contains only sodium silicate, apart from the excess of $\ce{NaOH}$. Silica can then be obtained pure by adding  enough $\ce{HCl}$ to destroy $\ce{Na2SiO3}$ and the NaOH in excess.$$\ce{Na2SiO3 + 2 HCl + H2O -&gt;  Si(OH)4 + 2 NaCl}$$ It is advisable to highly dilute the $\ce{NaOH + Na2SiO3}$ solution before adding $\ce{HCl}$. Because the precipitate of $\ce{Si(OH)4}$ is extremely voluminous, and fills up the entire solution.