Freaking geometric product

Hi, I just want to see if I understood this. Since the geometric product is associative and so on we can write for two multivectors A and B given by



[itex]A= \alpha_{0}+\alpha_{1}e_{1}+\alpha_{2}e_{2}+\alpha_{3}e_{1}\wedge e_{2}[/itex]

[itex]B= \beta_{0}+ \beta_{1}e_{1}+\beta_{2}e_{2}+\beta_{3}e_{1}\wedge e_{2}[/itex]



the geometric product multiplication as



[itex]AB=M=\mu_{0}+\mu_{1}e_{1}+\mu_{2}e_{2}+\mu_{3}e_{1}e_{2}[/itex]



Where for example [itex]\mu_{0}=\alpha_{0}\beta_{0}+\alpha_{1}\beta_{1}+\alpha_{2}\beta_{2}-\alpha_{3}\beta_{3}[/itex] and so on.



Now let's take an example with beautiful vectors with numbers like [itex]a = (e_{1}+2e_{2}), a_{1}=(2e_{1}+3e_{3}), a_{2}=(2e_{1}+0e_{2})[/itex]



So [itex]aa_{1}= 8- e_1\wedge e_{2}[/itex] Now what if I multiply this with [itex]a_{2}[/itex]?



[itex](8-e_{1}\wedge e_{2})(2e_{1})[/itex]? Is it just [itex]16e_{1}-(e_{1}\wedge e_{2})(2e_{1})[/itex] ? My logic behind this is that one can symbolically (GP) multiply multivectors and then opens up his list with like 300 products or something and then evaluates each of the products? Is this right?