I want to better understand the relationship between representations on dual vector spaces and opposite algebras when the algebra being represented is nonassociative. Specifically, my question is: does the dual space of a module over a nonassociative algebra induce a representation of the opposite algebra just like in the associative setting?
Here is what I understand: In the associative setting, when an algebra $\cal{A}$ is represented on a vector space $\cal{V}$ (i.e. we have $\pi:A\rightarrow End(\cal{V})$), which has a dual space $\cal{V}^\ast$, this naturally induces an opposite representation $\pi^o$:
$u( \pi(a)v) = (\pi^o(a)u)(v)$
where $u\in V^\ast$ and $v\in V$. The representation satisfies $\pi^o(a)\pi^o(b) = \pi^o(ba)$, so it is said to be a representation of the opposite algebra, where the product $\times$ on the opposite algebra satisfies $a\times b = ba$.
Ok, so here is my question. Nonassociative representations exist, but in general you can't write $\pi(a)\pi(b) = \pi(ab)$ (because the representation is non-associative). Is it nevertheless possible in general to induce a dual representation in a similar way on $V^\ast$, and show that it corresponds to a representation of the opposite algebra?
EXAMPLE: Let me use the octonion algebra as an example. The octonions have a natural bi-representation on themselves that satisfies properties like $L_{ab} = L_aL_b + [L_a,R_b]$, where $L_a$ and $R_a$ are the usual left and right actions of the algebra on itself. The opposite algebra has a representation on itself that satisfies e.g. $L^o_{ab} = L^o_bL^o_a + [L^o_b,R^o_a]$. The representation space is just $V=\mathbb{R}^8$, so if we take the opposite representation to be given by the transpose we find
$L^o_{ab} =L^T_{ab} = (L_aL_b + [R_a,L_b])^T = L^T_bL^T_a + [L^T_b,R^T_a] = L^o_bL^o_a + [L^o_b,R^o_a]$
where I have used the alternative property $[L_a,R_b]=[R_a,L_b]$. As 'expected' the representation on the dual space seems to reproduce the expected identity for the representation of the opposite algebra. But of course this is just 1 identity for one specific algebra - my question is more general: in general is the representation induced on the dual space also the representation of the opposite algebra?