While reading the definition of a representation of a Lie triple system, I have a few doubts which I have stated at the end.
Let $(T, \{\cdot,\cdot,\cdot\})$ be a Lie triple system and $V$ be a vector space along with a linear map $\theta: T \otimes T \to \mathrm{End}(V)$. Then $(V,\theta)$ is called a representation of $T$ if the following conditions are satisfied: For all $a,b,c,d \in T$ \begin{equation}\tag{R1} [D(a,b),\theta(c,d)] = \theta(\{a,b,c\},d) + \theta(c,\{a,b,d\}) \end{equation} \begin{equation}\label{R1} \tag{R2} \theta(a,\{b,c,d\}) = \theta(c,d) \theta(a,b) - \theta(b,d) \theta(a,c) + D(b,c) \theta(a,d) \end{equation} where $D(a,b) := \theta(a,b) - \theta(b,a)$, and $[D(a,b),\theta(c,d)]$ is the commutator of $D(a,b)$ and $\theta(c,d)$.
For example, there is a natural adjoint representation of $T$ on $T$ itself. The corresponding $D$ and $\theta$ are given by: \begin{equation*} D(a,b)(c) := \{a,b,c\} \quad \text{and} \quad \theta(a,b)(c) := \{c,a,b\}. \end{equation*}
My Questions:
In the case of adjoint representation of Lie triple system. How is $D(a,b) = \theta(a,b) - \theta(b,a)$ true?
While defining the representation of some algebra (Associative/Lie/Libniz/etc.) how does one arrive at the relations like (R1) and (R2) (in the above case)? What do these relations capture about the algebra?