$\langle \langle x + y, z \rangle \rangle = \langle \langle x, z \rangle \rangle + \langle \langle y, z \rangle \rangle$
It is clear when there are only $\langle \dot \ , \dot \ \rangle$ but what about the $\langle \langle \dot \ , \dot \ \rangle \rangle$
Please use the above mentioned notation.
I suspect that $\langle\langle\cdot,\cdot\rangle\rangle$ is used simply as a general scalar product, to distinguish from a particular scalar product (such as the dot product) that has been previously denoted by $\langle\cdot,\cdot\rangle.$