Geometric parametrization of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>S</mml:mi><mml:mi>O</mml:mi><mml:mo mathvariant="bold" stretchy="false">(</mml:mo><mml:mi>D</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo mathvariant="bold" stretchy="false">)</mml:mo></mml:math> phase space of all dimensional loop quantum gravity

To clarify the geometric information encoded in $SO(D+1)$ spin-network states for higher dimensional loop quantum gravity, we generalize twisted-geometry parametrization of $SU(2)$ phase space ($1+3$)-dimensional gravity to that all-dimensional case. The Poisson structure terms twisted variables suggests a new gauge reduction procedure, with respect discretized gauss and simplicity constraints governing kinematics theory. Endowed meaning via parametrization, our procedure serves identify proper freedom associated anomalous subsequently leads desired classical state (twisted) discrete Arnowitt-Deser-Misner data.