I do not recommend the oval or belgian cast for routine casting. The reason is line and leader twist.
By casting in an ellipse, you are moving the rod tip in a circle for each casting cycle of a backward and forward cast. For a right hander
this causes the fly line to twist counter clockwise
for each cycle. Take a look at how your fly line is twisted and I bet that you will find a counter clockwise twist for a right-hander and a clockwise twist for a left-hander.
To prove this to yourself, take a pencil in your like a rod and bring it back low to your side and then around the ellipse and forward high, then back low and forward high, over and over. Look at the circle your hand and arm are making and you will see that you are moving in a counter clockwise circle.
This type of elliptical cast is often used when casting in the wind, and is called the Belgian "wind" cast.
It is also used when nymphing when you pick up the line low to your side and then use a water tension cast to bring the cast back over your head. Then you follow the drift back and lower the rod to the side and loop over your head again. You are repeatedly casting elliptically.
To remove the twist, make a long cast downstream, and then opposite
clockwise loops with your fly line to remove the twists and then reel up the line onto the reel before leaving the stream.
Here's a video of the elliptical cast. You can see how he would introduce a twist with his casting motion.
"The disadvantage of this cast is that it throws a half twist in the line every cast. Half twists add up! So it's best to use this cast sparingly, otherwise you will have to get into the habit of removing the reel from the rod, every 30 casts, or so, in order to spin the twists out."
The Belgian Cast
The Belgian Cast | MidCurrent