I wanted the axis to be fixed at left, bottom corner of the screen and axis should rotate about it's own origin.

This variation leaves the axes at the origin, moves the box to

P = (size / 2, -size / 2, -size / 2)

relative to the axes, and pans the camera toward the bottom, center of the screen. Uncomment the call to camera.setTranslateX() to pan left. Moving the mouse causes the group to rotate about the axes' origin. Press shift to rotate about z, and use the mouse wheel to dolly the camera.

import javafx.application.Application;
import javafx.scene.Group;
import javafx.scene.PerspectiveCamera;
import javafx.scene.Scene;
import javafx.scene.input.MouseEvent;
import javafx.scene.input.ScrollEvent;
import javafx.scene.paint.Color;
import javafx.scene.paint.PhongMaterial;
import javafx.scene.shape.Box;
import javafx.scene.transform.Rotate;
import javafx.stage.Stage;

/**
 * @see http://***.com/a/37734966/230513
 * @see http://***.com/a/37714700/230513
 * @see http://***.com/a/37685167/230513
 * @see http://***.com/a/37370840/230513
 */
public class TriadBox extends Application {

    private static final double SIZE = 300;
    private final Content content = Content.create(SIZE);

    private static final class Content {

        private static final double WIDTH = 3;
        private final Group group = new Group();
        private final Rotate rx = new Rotate(0, Rotate.X_AXIS);
        private final Rotate ry = new Rotate(0, Rotate.Y_AXIS);
        private final Rotate rz = new Rotate(0, Rotate.Z_AXIS);
        private final Box xAxis;
        private final Box yAxis;
        private final Box zAxis;
        private final Box box;

        private static Content create(double size) {
            Content c = new Content(size);
            c.group.getChildren().addAll(c.box, c.xAxis, c.yAxis, c.zAxis);
            c.group.getTransforms().addAll(c.rz, c.ry, c.rx);
            return c;
        }

        private Content(double size) {
            xAxis = createBox(size * 2, WIDTH, WIDTH);
            yAxis = createBox(WIDTH, size * 2, WIDTH);
            zAxis = createBox(WIDTH, WIDTH, size * 2);
            double edge = 3 * size / 4;
            box = new Box(edge, edge, edge);
            box.setMaterial(new PhongMaterial(Color.CORAL));
            box.setTranslateX(size / 2);
            box.setTranslateY(-size / 2);
            box.setTranslateZ(-size / 2);
        }

        private Box createBox(double w, double h, double d) {
            Box b = new Box(w, h, d);
            b.setMaterial(new PhongMaterial(Color.AQUA));
            return b;
        }
    }

    @Override
    public void start(Stage primaryStage) throws Exception {
        primaryStage.setTitle("JavaFX 3D");
        Scene scene = new Scene(content.group, SIZE * 2, SIZE * 2, true);
        primaryStage.setScene(scene);
        scene.setFill(Color.BLACK);
        scene.setOnMouseMoved((final MouseEvent e) -> {
            if (e.isShiftDown()) {
                content.rz.setAngle(e.getSceneX() * 360 / scene.getWidth());
            } else {
                content.rx.setAngle(e.getSceneY() * 360 / scene.getHeight());
                content.ry.setAngle(e.getSceneX() * 360 / scene.getWidth());
            }
        });
        PerspectiveCamera camera = new PerspectiveCamera(true);
        camera.setFarClip(SIZE * 6);
        //camera.setTranslateX(SIZE / 2);
        camera.setTranslateY(-SIZE / 2);
        camera.setTranslateZ(-4.5 * SIZE);
        scene.setCamera(camera);
        scene.setOnScroll((final ScrollEvent e) -> {
            camera.setTranslateZ(camera.getTranslateZ() + e.getDeltaY());
        });
        primaryStage.show();
    }

    public static void main(String[] args) {
        launch(args);
    }
}