$\DeclareMathOperator\sym{sym}$Let $i$, $j$, $k$ be the units of quaternions, in particular $i^2=j^2=k^2=-1$, $ijk=-1$.

We will use non commutative variables $x$, $y$, $z$. Define $\sym_{a,b,c}$ to be the polynomial made of the sum of monomials which are all possible products of $a$ variables $x$, $b$ variables $y$ and $c$ variables $z$.

For example $\sym_{2,1,0}(i,j,k)=i^2 j+i j i+j i^2$.

Considering the symmetric definition of $\sym_{a,b,c}$ and the non commutativity of quaternions I would expect a lot of simplifications on $\sym_{a,b,c}(i,j,k)$, in fact $\sym_{a,0,0}(i,j,k)=i^a$, and $\sym_{1,1,1}(i,j,k)=0$ but $\sym_{2,2,0}(i,j,k)=i^2 j^2+j^2 i^2=2$.

Is there a simple formula to determine the value of $\sym_{a,b,c}(i,j,k)$ in terms of $a$, $b$, $c$?

May/29/2019 By the way I introduced those functions we have: $e^{(i+j+k)/\sqrt3}=\sum_n (\sqrt{3}^n/n!)\sum_{a,b,c} \sym_{a,b,c}(i,j,k) =(i+j+k)\sin(90)=i+j+k$.