I have a quaternion representation of rotation and position in a co-ordinate frame which is right-handed and I want the corresponding rotation and position in another frame where the Y-axis is flipped (so right-handed has become left-handed). Like in the figure:

For position I figured that I only have to negate the y coordinate, so that:

(x1, y1, z1) would become (x1, -y1, z1) in the new coordinate frame.

I'm not so sure about the quaternions. The way I think about it, isn't it sufficient that I negate the quaty1? So that only one rotation is negated? So if I have (quatw1, quatx1, quaty1, quatz1) is it sufficient to represent the same rotation in the new coordinate system as (quatw1, quatx1, -quaty1, quatz1)

There are multiple questions answered in Stack Overflow, but they somehow almost always start from how to go from left-hand to right-hand, and because quaternions inherently don't have a handedness, I am confused.

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