Rotation matrix to Euler angles (Static RPY) using the C++ Eigen library

TLDR:

How to get Euler angles in RPY form from a rotation matrix using the Eigen library. I tried ChatGPT, Google search and the documentation. The following is supposed to be the way: mat.eulerAngles(0, 1, 2). But, this does not seem to work.

Given a rotation matrix I want to get the Euler angles in the "sxyz" form. In other words I am trying to get ax, ay, az for a given rotation matrix such the the rotation matrix can be obtained as Rx * Ry * Rz, where each Ri is rotation about the axis i by the angle ai. I am using the C++ Eigen library version 3.7 for this.

I tried the following (please see code below) where I generate the matrix mat from Euler angles ax, ay, az by roll-pitch-yaw (RPY) rotation about static frame. I have verified that this matrix is correct. Following this I obtained the Euler angles (presumable in RPY format) from mat and used those euler angles to get mat2 by RPY rotation about static frame. I expect these two matrices to be same. But they are very different. What is wrong here?

#include <iostream>
#include <eigen3/Eigen/Dense>

int main() {
    // Initialize euler angles.
    double ax, ay, az;
    ax = M_PI / 2;
    ay = M_PI / 2;
    az = 0;

    // Quaternion from Euler angles.
    Eigen::Quaterniond q = Eigen::AngleAxisd(az, Eigen::Vector3d::UnitZ()) *
                Eigen::AngleAxisd(ay, Eigen::Vector3d::UnitY()) *
                Eigen::AngleAxisd(ax, Eigen::Vector3d::UnitX());
    
    Eigen::Matrix3d mat = q.toRotationMatrix();

    // Print rotation matrix to verify.
    std::cout << mat << std::endl;
    std::cout << "----" << std::endl;

    // Get Eigen angles from rotation matrix.
    Eigen::Vector3d ea = mat.eulerAngles(0, 1, 2);

    // Print euler angles.
    std::cout << ea << std::endl;
    std::cout << "----" << std::endl;
    
    Eigen::Quaterniond q2 = Eigen::AngleAxisd(ea(2), Eigen::Vector3d::UnitZ()) *
                Eigen::AngleAxisd(ea(1), Eigen::Vector3d::UnitY()) *
                Eigen::AngleAxisd(ea(0), Eigen::Vector3d::UnitX());
    
    Eigen::Matrix3d mat2 = q2.toRotationMatrix();

    // 
    std::cout << mat2 << std::endl;
    std::cout << "----" << std::endl;
}

The output is as follows:

2.22045e-16           1 2.22045e-16
          0 2.22045e-16          -1
         -1 2.22045e-16           0
----
     1.5708
2.22045e-16
    -1.5708
----
 3.33067e-16  2.77556e-16           -1
          -1            0 -2.77556e-16
-5.55112e-17            1  3.33067e-16

I think the Euler angles are obviously wrong. Also they do not generate the original rotation matrix. I have also tried alternatives like mat.eulerAngles(2, 1, 0). But none of them work.