Manipulate part's CFrame so that it moves along one axis while ignoring the rotation of another axis

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You can write your topic however you want, but you need to answer these questions:

  1. What do you want to achieve?
    I wish to change the position of the player’s camera by moving it in the direction of its local x-axis.

  2. What is the issue?
    When moving it in the direction of its local x-axis, it also applies the rotation the camera has on the z-axis, which I do not want. The camera should move independent of its pitch.

  3. What solutions have you tried so far?
    I tried experimenting with look vector and reading up a lot on CFrames, but I have no idea how to achieve this.

To better visualize my problem, assume this part is the camera:
image
This part has been rotated 45 degrees around the y-axis (green arrow) and 45 degrees around the z-axis (blue arrow)

If I were to move it along its x-axis, the red arrow, it would move downwards because I also rotated the part along the z-axis, the blue arrow.
What I want is to move it along the x-axis as if the z-axis was never transformed:
image
However, I do not want to actually change the z-axis rotation of the part, I just want to move it as if it was never changed.

This is what I have so far:

-- runs every render step, cam_pos is a global that describes position of camera,
-- x_angle describes the rotation of the x-axis, the pitch, which I want to ignore when mvoing forward
-- y_angle describes the rotation of the y-axis, the yaw, which determines the direction "forward" actually is
local function updateCamera(dt)
	if forward then
		-- this moves with the z-axis in mind
		cam_pos += camera.CFrame.LookVector
	end
	
	camera.CFrame = cam_pos * CFrame.fromOrientation(math.rad(x_angle), math.rad(y_angle), 0)
end

I would be happy to elaborate or provide more code samples if what I have now is too vague. Thanks for any help in advance.

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--All this is equivalent to `return cf * CFrame.new(translation)`, just for illustration purposes.
function translateCFrameRelativeToItself(cf: CFrame, translation: Vector3): CFrame
    local relativeTranslation = cf:VectorToObjectSpace(translation)
    return cf + relativeTranslation 
end
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