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The answer is yes, and you can see this by realizing that the vertical (resp. the horizontal) subbundle of $TTM$ is mapped to the vertical (resp. the horizontal) subbundle of $T^*TM.$ For the vertical part, this follows from the fact that $g\colon TM\to T^*M$ is a bundle homomorphism, and for the horizontal bundle this follows from the fact that $g\colon TM\to T^*M$ is parallel (with respect to the natural connections defined by $g$ which are used to define the horizontal bundles).