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Post Closed as "Not suitable for this site" by Jeremy Rickard, abx, YCor, Andreas Blass, Konstantinos Kanakoglou
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Is it possible(or may be easier) to give an example of non associative algebra but commutative?

At first sight, it seems possible to prove associativity from commutativity but later realised it may no be the case.

The only example of non associative algebra which I know is Octonion but which is non-commutative. Then I started to create an artificial example. Here is my approach(same as to create an commutative): Let's $\textbf{C}$ denote the category of associative algebra. Suppose free object exists in this category say $F$ be a free associative algebra, then consider the quotient of $F$ by the sub algebraideal generated by its commutator elements. I don't know how much this work but certainly this way we will have lot of such examples.

Is it possible(or may be easier) to give an example of non associative algebra but commutative?

At first sight, it seems possible to prove associativity from commutativity but later realised it may no be the case.

The only example of non associative algebra which I know is Octonion but which is non-commutative. Then I started to create an artificial example. Here is my approach(same as to create an commutative): Let's $\textbf{C}$ denote the category of associative algebra. Suppose free object exists in this category say $F$ be a free associative algebra, then consider the quotient of $F$ by the sub algebra generated by its commutator elements. I don't know how much this work but certainly this way we will have lot of such examples.

Is it possible(or may be easier) to give an example of non associative algebra but commutative?

At first sight, it seems possible to prove associativity from commutativity but later realised it may no be the case.

The only example of non associative algebra which I know is Octonion but which is non-commutative. Then I started to create an artificial example. Here is my approach(same as to create an commutative): Let's $\textbf{C}$ denote the category of associative algebra. Suppose free object exists in this category say $F$ be a free associative algebra, then consider the quotient of $F$ by the ideal generated by its commutator elements. I don't know how much this work but certainly this way we will have lot of such examples.

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non-sssociativeassociative but commutative algebra

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non-sssociative but commutative algebra

Is it possible(or may be easier) to give an example of non associative algebra but commutative?

At first sight, it seems possible to prove associativity from commutativity but later realised it may no be the case.

The only example of non associative algebra which I know is Octonion but which is non-commutative. Then I started to create an artificial example. Here is my approach(same as to create an commutative): Let's $\textbf{C}$ denote the category of associative algebra. Suppose free object exists in this category say $F$ be a free associative algebra, then consider the quotient of $F$ by the sub algebra generated by its commutator elements. I don't know how much this work but certainly this way we will have lot of such examples.