nucleus

Let $A$ be an algebra, not necessarily associative multiplicatively. The nucleus of $A$ is:

$$𝒩(A):=\{a\in A\mid [a,A,A]=[A,a,A]=[A,A,a]=0\},$$

where $[,,]$ is the associator bracket. In other words, the nucleus is the set of elements that multiplicatively associate with all elements of $A$. An element $a\in A$ is nuclear if $a\in 𝒩(A)$.

$𝒩(A)$ is a Jordan subalgebra of $A$. To see this, let $a,b\in 𝒩(A)$. Then for any $c,d\in A$,

$[ab,c,d]$$=$$((ab)c)d-(ab)(cd)=(a(bc))d-(ab)(cd)$(1)

$=$$a((bc)d)-(ab)(cd)=a(b(cd))-(ab)(cd)$(2)

$=$$a(b(cd))-a(b(cd))=0$(3)

Similarly, $[c,ab,d]=[c,d,ab]=0$ and so $ab\in 𝒩(A)$.

Accompanying the concept of a nucleus is that of the center of a nonassociative algebra $A$ (which is slightly different from the definition of the center of an associative algebra):

$$𝒵(A):=\{a\in 𝒩(A)\mid [a,A]=0\},$$

where $[,]$ is the commutator bracket.

Hence elements in $𝒵(A)$ commute as well as associate with all elements of $A$. Like the nucleus, the center of $A$ is also a Jordan subalgebra of $A$.