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I am interested in clearing up a confusion of mine. I will try to make my question as clear as I can but I apologize in advance if this is not the case.

Given a unitary group of some unital involutive algebra, $U(A) =$ {$u \in A| uu^* = u^*u = I$},

I can define the adjoint action (and right action shortly) of u as $(Ad u)\xi:=u\xi u^*$, Where $\xi \in H$, the Hilbert space on which I am representing the algebra $A$.

Define right action on elements of the hilbert space using Connes Real Structure operator J. $\xi u^* = JuJ^*\xi$. At the level of the algebra the adjoint is then given as $(ad B)\xi:= B\xi - \xi B$.

My Question: How does this generalize in the case where the algebra A is non-associative and H is a general 'bimodule' over A? is conjugation always well defined?

For example. Say I am interested in the group G2 which is generated by the derivation algebra of the octonions (ie it is the automorphism group of the octonions). I can write out the root diagram for G2 so I know there should be an adjoint representation. The octonions O however are are not-associative. As further example if I take A = H = O, how is the adjoint action and conjugation defined (explicitly)?

I imagine there is a very simple answer to this question. Any direct help you could offer or an appropriate reference would be greatly appreciated.

2 added 117 characters in body

I am interested in clearing up a confusion of mine. I will try to make my question as clear as I can but I apologize in advance if this is not the case.

Given a unitary group of some unital involutive algebra, $U(A) =$ {$u \in A| uu^* = u^*u = I$},

I can define the adjoint action of u as $(Ad u)\xi:=u\xi u^*$, Where $\xi \in H$, the Hilbert space on which I am representing the algebra $A$.

Define right action on elements of the hilbert space using Connes Real Structure operator J. $\xi u^* = JuJ^*\xi$. At the level of the algebra the adjoint is then given as $(ad B)\xi:= B\xi - \xi B$.

My Question: How does this generalize in the case where the algebra A is non-associative and H is a general '(bi?)module' bimodule' over A? is conjugation always well defined?

For example. Say I am interested in the group G2 which is generated by the derivation algebra of the octonions (ie it is the automorphism group of the octonions). I can write out the root diagram for G2 so I know there should be an adjoint representation. The octonions O however are are not-associative. As further example if I take A = H = O, how is the adjoint action and conjugation defined (explicitly)?

I imagine there is a very simple answer to this question. Any direct help you could offer or an appropriate reference would be greatly appreciated.

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# Reference request (or otherwise): Adjoint action

I am interested in clearing up a confusion of mine. I will try to make my question as clear as I can but I apologize in advance if this is not the case.

Given a unitary group of some unital involutive algebra, $U(A) =$ {$u \in A| uu^* = u^*u = I$},

I can define the adjoint action of u as $(Ad u)\xi:=u\xi u^*$, Where $\xi \in H$, the Hilbert space on which I am representing the algebra $A$. At the level of the algebra the adjoint is then given as $(ad B)\xi:= B\xi - \xi B$.

My Question: How does this generalize in the case where the algebra A is non-associative and H is a general '(bi?)module' over A? is conjugation always well defined?

For example. Say I am interested in the group G2 which is generated by the derivation algebra of the octonions (ie it is the automorphism group of the octonions). I can write out the root diagram for G2 so I know there should be an adjoint representation. The octonions O however are are not-associative. As further example if I take A = H = O, how is the adjoint action and conjugation defined (explicitly)?

I imagine there is a very simple answer to this question. Any direct help you could offer or an appropriate reference would be greatly appreciated.